{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0001","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22b9aa23ec78e5e343f592ae07b2b254858e29a46278783a32dce105b38a988a","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257001,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-12,-2,-3,1,-1,-2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0002","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"858456a256aa8833f2eee16c8f1c21b4ac1493004e8f5ca2b52e4abe116d367e","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257002,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-11,-1,-2,2,0,-1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0003","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f82e11087f300174947e6ee76c425b46ba8d8009df4fa50fe065d89a08e9db4","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257003,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-10,0,-1,3,1,0]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [0, 6, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [0, 6, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0004","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d82246cf65cde9f7680ea5880e2a7aa81065dcccd6f9e55b16a469aa7e20a0cf","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257004,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-9,1,0,4,-1,1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0005","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ed424c561a07d5a2c4ee41298d417692e5a6b5acdba3acbb51560256e4c0615","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257005,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-8,2,1,1,0,2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [2, 2, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [2, 2, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0006","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f334fa0749cd664f14907d6d66e27de2e51b39023b45182ecd2faaae4e7dd86b","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257006,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-7,-2,2,2,1,-2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [-2, 4, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [-2, 4, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0007","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"518639b5f3040a49fdbb1268a1e0eb0a7e7abf7fa9f57e79024fc079d984f16c","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257007,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-6,-1,3,3,-1,-1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [-1, 6, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [-1, 6, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0008","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bd191907a6a9f9b68f534ed9aa0089b3275879cb3273c2ce14db327e7ff6356","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257008,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-5,0,-3,4,0,0]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0009","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a7b6e73dfca3bec3715a00741bce20290afb870b81be483de41588c5f211c7a","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257009,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [1, 2, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [1, 2, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0010","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1740880dd2a6da32e1a1b9b8f39e41f99c020ac7eb570a4c013f19f1dcae548c","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257010,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-3,2,-1,2,-1,2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [2, 4, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [2, 4, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0011","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0a93d9fdd5dad934a203da3eff693d11a93464f42f62ea0bda4af9caca9f017","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257011,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-2,-2,0,3,0,-2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [-2, 6, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [-2, 6, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0012","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c68a92b7dd5500971a80d49c64268f01ab1d1b99a7b9bafeb96d655e2c888903","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257012,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-1,-1,1,4,1,-1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0013","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6618ed6697f63f9673a8b70b498aeacb5b4363a7a14c0b5a56f6de54fdcf4e4","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257013,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[0,0,2,1,-1,0]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [0, 2, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [0, 2, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0014","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e18845297245583c21d108309db3a5681b4471b918f558d5b5e9120ecedb74b2","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257014,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[1,1,3,2,0,1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [1, 4, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [1, 4, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0015","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46139cc036a1b8e8faf0c114625284eaad4c7aafd975477a2e82c728ab2dff0e","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257015,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[2,2,-3,3,1,2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [2, 6, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [2, 6, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0016","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61dbc114cd6c7b5895e9f1ffcf5ba306a4412814647a374b2d40badc0a5787a3","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257016,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[3,-2,-2,4,-1,-2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [-2, 8, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [-2, 8, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0017","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8060a37aa3a5eff1f9197dc98adcb90fb166b00e71184e80cb716203c9c74c95","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257017,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[4,-1,-1,1,0,-1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [-1, 2, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [-1, 2, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0018","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12ce10cfc2d121f48c07699efc0a8bcf5a84ff939882561eb2cf36dc556a9101","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257018,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[5,0,0,2,1,0]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0019","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f76b8630491963123d390e42e3597ab7cba956f53b3d12a21276c0f393839c","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257019,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[6,1,1,3,-1,1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, 6, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, 6, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0020","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7eadd2b41526f3b2519674022529b38addad186e3f7d8c0cf37c66b66c36b7c","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257020,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[7,2,2,4,0,2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0021","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3453d46028e9f2eef13c076acca0416a389f568ddee169a1eb78d5f68872e1c0","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257021,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[8,-2,3,1,1,-2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [-2, 2, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [-2, 2, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0022","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65f2ac5e965f2c711d28d41adf04266499337228dfbdfff79e4e530861392b47","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257022,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[9,-1,-3,2,-1,-1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0023","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"789b310bdd46c438c413fddc836f9661df22807f37c8089ff463e20a6fc9ddcd","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257023,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[10,0,-2,3,0,0]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [0, 6, 0], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [0, 6, 0], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0024","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92520ec1759e98f225ca954cc7e1073af75a98f464531a7cdfba6dadefc070ca","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257024,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[11,1,-1,4,1,1]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0025","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2d6403495736bbe792a3517ceaeb1b06e5141dc88a0a876f01db8275ae0faf5","family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257025,"semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[12,2,0,1,-1,2]}"},"fields":{"title":"The Entire x-Partial Polynomial in an Explicit Basis","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [2, 2, -1], [1,x,y])","explanation":"Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [2, 2, -1], [1,x,y])."},"skills":["The Entire x-Partial Polynomial in an Explicit Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0026","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34110a49695d136ad64b0aa17260be3c70fadf342206b9347fc029c4ae51ad5c","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257026,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-12,-2,-3,1,-1,-2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0027","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82976e8e2037c534fe577a3bcfe25c6028bd5054d6974096f5e8bcdf1ede0708","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257027,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-11,-1,-2,2,0,-1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [-2, 0, -2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [-2, 0, -2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0028","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5570e967b1c18e00ccf87b27b37f8ad62c41302f81755e989007222c7354a091","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257028,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-10,0,-1,3,1,0]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0029","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02aaf9e5681ef834fac68ed3d0448fb4677596e8948ffb0adda29bbef9a44468","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257029,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-9,1,0,4,-1,1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0030","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55df44f81e2cf8e40ec32c58335925020a8f2bab4d1edb533d345f69c7ef75ee","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257030,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-8,2,1,1,0,2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [1, 0, 4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [1, 0, 4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0031","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"393ced48e7401af3637a86a2291d3c613699c0c3379e0a3d99aef25e53c1bdbe","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257031,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-7,-2,2,2,1,-2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [2, 1, -4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [2, 1, -4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0032","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c239d53bc7757b3bda63891974ef798432dac0d86a4b2a671a5f31fb863d799","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257032,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-6,-1,3,3,-1,-1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [3, -1, -2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [3, -1, -2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0033","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f78c81da0fcce17d1666078aedcbb7b943b203bce93fbbaecd1489beed3b3fb6","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257033,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-5,0,-3,4,0,0]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0034","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42f1e10101ffc9456e792b6fca9ec642cba86280edee8c1282f3c471122581dc","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257034,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0035","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa308caca3dd0164b23ae5ec6b78df2873e559bae73a82e646f9ac6b3c41b0e3","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257035,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-3,2,-1,2,-1,2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [-1, -1, 4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [-1, -1, 4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0036","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9256231a8bfa51c688243b68bb90691ea37caab2c443c4852e772120fbc514fa","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257036,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-2,-2,0,3,0,-2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [0, 0, -4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [0, 0, -4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0037","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff017123da4b9c8bfa6506faf282217360526feda9895c8c06e99f022bdf3524","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257037,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-1,-1,1,4,1,-1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [1, 1, -2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [1, 1, -2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0038","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43c086b6c6a5fbc43b50ccc141a9bff6327d6990a17ef19b769a923ac5de685d","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257038,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[0,0,2,1,-1,0]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [2, -1, 0], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [2, -1, 0], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0039","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6ebd3ccafe730654c76e1b8daaf9098fbc18a3a69f8d18f8ce784421ca3315c","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257039,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[1,1,3,2,0,1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [3, 0, 2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [3, 0, 2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0040","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14de4e4dbac44cf2b191d26701374c0df96e4f7ee0ac35a3b5fefbd8d5d3a1e7","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257040,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[2,2,-3,3,1,2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [-3, 1, 4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [-3, 1, 4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0041","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e5907bc2839538dfae7d4a545757b4dc6cdc041e7ff7080468c9b8444bb3dd8","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257041,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[3,-2,-2,4,-1,-2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [-2, -1, -4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [-2, -1, -4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0042","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9970226580828c7cb019270433b8600c09d9c1b0c69b152ee186fc9c77ef53f9","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257042,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[4,-1,-1,1,0,-1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [-1, 0, -2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [-1, 0, -2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0043","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0792b6603befea9208766f6633e6e6c530ffd5624a3696ab9dd89e63961c3b","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257043,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[5,0,0,2,1,0]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [0, 1, 0], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [0, 1, 0], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0044","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ca9d150f06b7ab913b8e8e611ecee5c83646ff8b432a3f160dbf4f72ff1bb6","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257044,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[6,1,1,3,-1,1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, -1, 2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, -1, 2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0045","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb9ac4589ff2109f9fb5960af05aabb8aebf4b7bf94bb30ca9663596209f35fe","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257045,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[7,2,2,4,0,2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0046","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef102f9ae1f068a2dfd4371c0ada4fd80f9e8a2eda21fd75abe2f3648761f4f","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257046,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[8,-2,3,1,1,-2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [3, 1, -4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [3, 1, -4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0047","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec3e2639d7c390a44e46dbdcd0a7ef38c98de3f4b12ee0baeadad5a18d95e3ec","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257047,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[9,-1,-3,2,-1,-1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [-3, -1, -2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [-3, -1, -2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0048","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eed28d500f9999b9ae8269618f1cfc9311534e6ae4795c43a877bbfc82bb200","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257048,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[10,0,-2,3,0,0]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [-2, 0, 0], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [-2, 0, 0], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0049","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"787a0731668fcc1f4a74a47505765d373d67315210275a6e31b05b912458cd75","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257049,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[11,1,-1,4,1,1]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0050","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bec0a036ac0eb06b76bce396c10488b15384d8cac98da96085f6e017126e0624","family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257050,"semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[12,2,0,1,-1,2]}"},"fields":{"title":"The Entire y-Partial Polynomial in the Same Coordinate Basis","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y])","explanation":"The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y])."},"skills":["The Entire y-Partial Polynomial in the Same Coordinate Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0051","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa507a0c511a892031fed23d21dc5ff403a3b3be94d39f832d83cd871382c8a2","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257051,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"fixed_x\":-2,\"y\":-3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0052","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"203dadc20b6a6b947f95f13127dd0d254cc915787035905aabfd857777276fc8","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257052,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-11,-1,-2,2,0,-1],\"fixed_x\":-1,\"y\":-2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], -1, [-8, -2, -1], [-2, -2], -2, 2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], -1, [-8, -2, -1], [-2, -2], -2, 2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0053","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2320648a76d6b6d6fb02b58b9e4c38e504cdb48b0669245b07553562cb9315a","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257053,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-10,0,-1,3,1,0],\"fixed_x\":0,\"y\":-1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 0, [-10, -1, 0], [-1, 0], -1, -1)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], 0, [-10, -1, 0], [-1, 0], -1, -1)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0054","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4680b0db792132b6be229595a95c8742d40b9d254cc1df5ac8bd36e2cf589dd","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257054,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-9,1,0,4,-1,1],\"fixed_x\":1,\"y\":0}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 0, -1)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 0, -1)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0055","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05b7e0a8a6947c1c1b267311e261863fdc5aa35fe5b9b1c8696839af03eb2461","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257055,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-8,2,1,1,0,2],\"fixed_x\":2,\"y\":1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 5)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 5)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0056","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0045e4fe0f5e2af2d988998bffc7b59369db770d34057ba1f75ea8d6d25ab1f2","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257056,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-7,-2,2,2,1,-2],\"fixed_x\":-2,\"y\":2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0057","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fa616924ce3228e95481e0b487991f7030d1c539dc49d98cb4adc5d2ceeae4c","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257057,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-6,-1,3,3,-1,-1],\"fixed_x\":-1,\"y\":3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], -1, [-2, 4, -1], [4, -2], 3, -2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], -1, [-2, 4, -1], [4, -2], 3, -2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0058","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f855c8a6386852cce0cfe2867b72f9df869e857ec4aac0c3da293c01c0b0e36","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257058,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-5,0,-3,4,0,0],\"fixed_x\":0,\"y\":-3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 0], -3, -3)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 0], -3, -3)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0059","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c82eaf754d0836a2661c8ec52ccdb573e129cc237d0fd5a0e4c79524accf1878","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257059,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-4,1,-2,1,1,1],\"fixed_x\":1,\"y\":-2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 1, [-2, -1, 1], [-1, 2], -2, -5)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], 1, [-2, -1, 1], [-1, 2], -2, -5)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0060","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeeaac99dd63fbe29696b6f9c15a7d09acab6e9b04f55ba9e6438414ae237b52","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257060,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-3,2,-1,2,-1,2],\"fixed_x\":2,\"y\":-1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -1, -7)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -1, -7)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0061","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fca241ac36c5e9d3a7b0e60db2867aa3ab21a2fd2668043a011e2adf80b49a76","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257061,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-2,-2,0,3,0,-2],\"fixed_x\":-2,\"y\":0}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 0, 0)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 0, 0)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0062","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f1791e3b157d2a7bb0fd55f74b122c2c088eac0298024ce1c4f567ee57852a8","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257062,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-1,-1,1,4,1,-1],\"fixed_x\":-1,\"y\":1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], -1, [4, 0, -1], [0, -2], 1, -2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], -1, [4, 0, -1], [0, -2], 1, -2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0063","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79456bb8f73c86a1b8d18373e75cb231ed6c2b328513bea8cfc516fc8bb4a674","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257063,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[0,0,2,1,-1,0],\"fixed_x\":0,\"y\":2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 0, [0, 2, 0], [2, 0], 2, 2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], 0, [0, 2, 0], [2, 0], 2, 2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0064","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"418f1a6fe0ec6122590938c0fa23144294d3899667ba9de1c83489b794321e66","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257064,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[1,1,3,2,0,1],\"fixed_x\":1,\"y\":3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0065","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aa99b85d2bec084a04e9113dadb0503c82f34b253d60e99c044eb47d1f57823","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257065,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[2,2,-3,3,1,2],\"fixed_x\":2,\"y\":-3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0066","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271cb2a815f7b6c37667b092d7364b531089f693cd103fa0b4b060c96fbe75b1","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257066,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[3,-2,-2,4,-1,-2],\"fixed_x\":-2,\"y\":-2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0067","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bfc3143cb53d7fa1e83ef6eb362a6adfa93ef064701906bb5bc50344e0552b1","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257067,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_x\":-1,\"y\":-1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, [6, -1, -1], [-1, -2], -1, 1)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [6, -1, -1], [-1, -2], -1, 1)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0068","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9224868f2f9beb123848c0fdeea4ff7b3972716620d2b16f493c58f74d062353","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257068,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[5,0,0,2,1,0],\"fixed_x\":0,\"y\":0}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [0, 0], 0, 0)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [0, 0], 0, 0)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0069","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ec97a6d4e242b3140d62d3f2e63147e6f796f273a7173705fba5e296e14536","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257069,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[6,1,1,3,-1,1],\"fixed_x\":1,\"y\":1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 1, [10, 0, 1], [0, 2], 1, 2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], 1, [10, 0, 1], [0, 2], 1, 2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0070","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8871886eae12d10a4eb1c34b62e176bc4415c844d791e8b5b44bef1365ad8262","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257070,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[7,2,2,4,0,2],\"fixed_x\":2,\"y\":2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0071","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f82a263d7b4b0c32b61c112b24dfd1e4429a8f7db516efd220f9b88aad99a095","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257071,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[8,-2,3,1,1,-2],\"fixed_x\":-2,\"y\":3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0072","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b526641b28afd652df7e1404abceb1e8dbf66cadc606d47adbc6f947eef32602","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257072,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0073","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e748e077a2050f2c58572f90ba2a5fddb547549fba1b11d56f884173c145b540","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257073,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[10,0,-2,3,0,0],\"fixed_x\":0,\"y\":-2}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 0], -2, -2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 0], -2, -2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0074","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e32c430f462e1539e989aa8c032dc996b1d0565fe3ff70be97693673343a4b45","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257074,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[11,1,-1,4,1,1],\"fixed_x\":1,\"y\":-1}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 1, [16, 0, 1], [0, 2], -1, -2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], 1, [16, 0, 1], [0, 2], -1, -2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0075","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"965c19fe74851bf286e8d41395000177b22550d8b8791990c0f1a0be3ecbf1eb","family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257075,"semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[12,2,0,1,-1,2],\"fixed_x\":2,\"y\":0}"},"fields":{"title":"A Fixed-x Slice and Its y-Derivative","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 0, -2)","explanation":"Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 0, -2)."},"skills":["A Fixed-x Slice and Its y-Derivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0076","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"200f3e0bd14ea5f3dacc78ef6edd9689a40c6f7e565f5282c8d2aa7cc2f311ec","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257076,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"x\":-2,\"y\":-3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0077","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b5043f8efc617f7a329e9d620095e9c7716dc6b6335207a690c27a885eba4e9","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257077,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-11,-1,-2,2,0,-1],\"x\":-1,\"y\":-2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0078","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61f307c6e13c8a471fb4736f2df959ccb7255c93875988aea811a3c5227bad29","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257078,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-10,0,-1,3,1,0],\"x\":0,\"y\":-1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0079","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"718cded7a514abf80c5146f62f9a8b248936ce87c914dab6490db938b92e948a","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257079,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-9,1,0,4,-1,1],\"x\":1,\"y\":0}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0080","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eee1b403be387bc86742105485b58ed653b7748a706047f5aaeff64e6609fc3","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257080,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-8,2,1,1,0,2],\"x\":2,\"y\":1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0081","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51efbe7c58fd9c91f3621985d0231c78730b96efa8a28e1b8f42867f7f62956d","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257081,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-7,-2,2,2,1,-2],\"x\":-2,\"y\":2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0082","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4d2b88aa833bb840e5fca3087caef5494cd6095ecd50a7d792864b3949f294c","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257082,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-6,-1,3,3,-1,-1],\"x\":-1,\"y\":3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0083","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59545e48b5fd2139c9470f7799ad447f0deb0a0e683e4e99b9fc0bfd636e5089","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257083,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-5,0,-3,4,0,0],\"x\":0,\"y\":-3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0084","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782090fbb551dd58a5a000902013c127d56e4666bc8d78490dc035d6ef5dd6b3","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257084,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-4,1,-2,1,1,1],\"x\":1,\"y\":-2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0085","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"895f7fb1d29ab29c103144cc31823601c131e3aabf35c56269c2ad8ddf5ebec1","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257085,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-3,2,-1,2,-1,2],\"x\":2,\"y\":-1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0086","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b4a484f4de8e6788b73562e9825a5256113b1ab2892ea18c81d093102749d0d","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257086,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-2,-2,0,3,0,-2],\"x\":-2,\"y\":0}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0087","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89bfb7327c3a844ea30a21700a0b48943a5d8e7d1260c8a467abbab341662c53","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257087,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-1,-1,1,4,1,-1],\"x\":-1,\"y\":1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0088","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"170a6b2d21610c23d9926a6eb0363e86b1a256ef0ae3ca25ee91890ae748fc13","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257088,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[0,0,2,1,-1,0],\"x\":0,\"y\":2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0089","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84c397cdbb3594306df8fdc2c02e25f590b8700394e8e1715b99e2ae6c051bfd","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257089,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[1,1,3,2,0,1],\"x\":1,\"y\":3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0090","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"862d30c19bbaba8cbf7a5eccd0106ccc828254c164642001a309ae8a542d5821","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257090,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[2,2,-3,3,1,2],\"x\":2,\"y\":-3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0091","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4918758c80bfe86c00f46a5e3f0981bd8312a43fd3cb083a2a45908e190f7fe","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257091,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[3,-2,-2,4,-1,-2],\"x\":-2,\"y\":-2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0092","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46305296119aa47f9b0457936aeb7108cb6ec7b80d6fa92dc38569f1b0098156","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257092,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0093","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"516913fedee1e83b6c181abc54b251cef43f3d7b258f80e750a8e0a35fc52829","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257093,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[5,0,0,2,1,0],\"x\":0,\"y\":0}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0094","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43cab1c2f490b02a6b2f7de91c388062e4ae045fe8cd4a33d66193c061ead62f","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257094,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[6,1,1,3,-1,1],\"x\":1,\"y\":1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0095","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d8a449c85dfa1b228e0dc4873967f4fbff6d0538c68ecc16759c8d74030263","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257095,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[7,2,2,4,0,2],\"x\":2,\"y\":2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0096","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec29c76c7308b69d9a77409404f79e7f9a63c6f834eced83662a6afeade5a052","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257096,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[8,-2,3,1,1,-2],\"x\":-2,\"y\":3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0097","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb18d89b46462f34231460b5549c596af84c6aa692e3ce2d5acb6d5a0fdf0713","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257097,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[9,-1,-3,2,-1,-1],\"x\":-1,\"y\":-3}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0098","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"869d7882b5601e201d63d5d67fcfcbe2f03843ffe71f74a144c0776edcb22cbc","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257098,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[10,0,-2,3,0,0],\"x\":0,\"y\":-2}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0099","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada64cd32311fbffedc81180bc19cd0e31b567246bc1445e3ad768dd04886815","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257099,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[11,1,-1,4,1,1],\"x\":1,\"y\":-1}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r07-partial-derivative-0100","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3ae634b95d116fd24f8291de86472497800ce29304d6e8272c3e6512c190efa","family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257100,"semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[12,2,0,1,-1,2],\"x\":2,\"y\":0}"},"fields":{"title":"Cancel Only Nonzero h before Taking the Symbolic Slice Limit","stem":"Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation":"Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)."},"skills":["Cancel Only Nonzero h before Taking the Symbolic Slice Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}