{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r05-derivative-rules-0001","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f10ec2336850d8ef83bd920016f6537b863f5cba2fdf72565d3c7560dcbce9e","family_id":"qf-derivative-rule-sum","seed":202610085001,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[\"constant_sum\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [3, 1], starting with its constant term; g has ascending coefficients [-5, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use f′+g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0002","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7398e9d38b31e806ad821c8fcfc4f04ff1af650c00c9d0865810c0816c5bac8","family_id":"qf-derivative-rule-sum","seed":202610085002,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"1\",\"1\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"8","explanation":"Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0003","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788ef131fe5a5652ff2f52a2506317df96c5fec60725d90110a73c418112dd9b","family_id":"qf-derivative-rule-sum","seed":202610085003,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"2\",\"2\",\"2\"],\"0\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [0, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"2","explanation":"Use f′+g′. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0004","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e79a7947811cbd0b62bfca8c4459d2ee6d6a34492d4ee2d664fa35138925e9b1","family_id":"qf-derivative-rule-sum","seed":202610085004,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"3\",\"0\",\"5\"],\"1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [-1, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"18","explanation":"Use f′+g′. The exact result is 18; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0005","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa4098e0062361971c27d25315dc1b2b8d4c85cf897cbc14bac0dc88c4911edf","family_id":"qf-derivative-rule-sum","seed":202610085005,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"4\",\"1\"],\"2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [-2, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"8","explanation":"Use f′+g′. The exact result is 8; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0006","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2281f59a768662049d09e5cc941043b08b6869f00cbbed121fe385b120b0b131","family_id":"qf-derivative-rule-sum","seed":202610085006,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"5\",\"2\",\"3\"],\"-2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [-3, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"33","explanation":"Use f′+g′. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0007","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9ac4a1c77b4f783754bad67f92f24d536e7433389cb067f2e4761d421bd0371","family_id":"qf-derivative-rule-sum","seed":202610085007,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"6\",\"0\",\"2\"],\"-1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [-4, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"12","explanation":"Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0008","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ecae8305dbee3266aabc7daf246df648174867c28f27873693764e42dde5ef","family_id":"qf-derivative-rule-sum","seed":202610085008,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"7\",\"1\",\"5\"],\"0\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [-5, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"7","explanation":"Use f′+g′. The exact result is 7; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0009","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbb463775fd3b03ef2a9534fc2a3cb034d548d057e04c30a274208d1b22dbd4b","family_id":"qf-derivative-rule-sum","seed":202610085009,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"8\",\"2\"],\"1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [-6, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"12","explanation":"Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0010","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76496dcaf01e94f510ecc1190f69fbf15508b48f3f3ad9ad7efe9c682210ba6a","family_id":"qf-derivative-rule-sum","seed":202610085010,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"9\",\"0\",\"3\"],\"2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [-7, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"45","explanation":"Use f′+g′. The exact result is 45; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0011","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31b3c8ee03ffe18767a41307b303a26a41649054c9be6756b8e9ff257edcdfb5","family_id":"qf-derivative-rule-sum","seed":202610085011,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"10\",\"1\",\"2\"],\"-2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [-8, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"30","explanation":"Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0012","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0aba44814feda563a759397bae630f1c32fdf7d2a86aef59521b6092c9bb0bf","family_id":"qf-derivative-rule-sum","seed":202610085012,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"11\",\"2\",\"5\"],\"-1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [-9, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"22","explanation":"Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0013","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3182f0bb0da2162b862a32b7ba392830689b42ade91d713ef4c5a852e594357a","family_id":"qf-derivative-rule-sum","seed":202610085013,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"12\"],\"point_irrelevant\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [-10, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"12","explanation":"Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0014","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ded85e11e4125612191911eb5f64bd50474810f25bd7f4ad8fbf910944ec3366","family_id":"qf-derivative-rule-sum","seed":202610085014,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"13\",\"1\",\"3\"],\"1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [-11, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"24","explanation":"Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0015","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebbeb38dba922116d8ae364826f06823dbfe980d133a69b39a0d2f02ee42edd7","family_id":"qf-derivative-rule-sum","seed":202610085015,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"14\",\"2\",\"2\"],\"2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [-12, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"46","explanation":"Use f′+g′. The exact result is 46; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0016","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7d266aecbadde87262e7f298e1b4fc8dda5ff9e75c802c3bfe121bc9907f56c","family_id":"qf-derivative-rule-sum","seed":202610085016,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"15\",\"0\",\"5\"],\"-2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [-13, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"75","explanation":"Use f′+g′. The exact result is 75; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0017","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4122d596c0491071c5ccaa04b2d068da0bb1a202caee96d6e39dd3cb7820bd71","family_id":"qf-derivative-rule-sum","seed":202610085017,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"16\",\"1\"],\"-1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [-14, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"14","explanation":"Use f′+g′. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0018","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb470500c27bb46023c1a189ae596efe18522e07dddf7be7504f2f826f093ca6","family_id":"qf-derivative-rule-sum","seed":202610085018,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"17\",\"2\",\"3\"],\"0\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [-15, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"17","explanation":"Use f′+g′. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0019","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb3ef24588a44b351c40ef78f49cca894fc390dbfd824eb65e613ca3323dd09b","family_id":"qf-derivative-rule-sum","seed":202610085019,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"18\",\"0\",\"2\"],\"1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [-16, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"24","explanation":"Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0020","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5a3e35f32d2d64a49cb8c02bf3b7e92c42860bc00a8a22029f8bb2a05f19c6f","family_id":"qf-derivative-rule-sum","seed":202610085020,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"19\",\"1\",\"5\"],\"2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [-17, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"83","explanation":"Use f′+g′. The exact result is 83; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0021","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f733df0d2a02df45d225ed612d48712b767c7b6d4daf2dbfde724c10a843ed","family_id":"qf-derivative-rule-sum","seed":202610085021,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"20\",\"2\"],\"-2\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [-18, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"12","explanation":"Use f′+g′. The exact result is 12; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0022","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ecb49ee6924608fe660611d408a83bdf154c88906a77200ee8f582dbd5b3ab7","family_id":"qf-derivative-rule-sum","seed":202610085022,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"21\",\"0\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [-19, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"30","explanation":"Use f′+g′. The exact result is 30; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0023","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83e4773488f52ca6791e12cc1e78fbb9a2a72103df18ad92bbd25e8c1ac9beb","family_id":"qf-derivative-rule-sum","seed":202610085023,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"22\",\"1\",\"2\"],\"0\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [-20, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"22","explanation":"Use f′+g′. The exact result is 22; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0024","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1260f5b1e8e344aa63a76d92f49ac61970153f656647b9f5f2b6d2ee8be8c3ca","family_id":"qf-derivative-rule-sum","seed":202610085024,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"23\",\"2\",\"5\"],\"1\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [-21, -1, 1, 1], starting with its constant term. Find the derivative of f+g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"42","explanation":"Use f′+g′. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0025","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0bda1923d9cf32e2a5d4518a53afc23ad6955095604018ed4ef23e3a4fc12d0","family_id":"qf-derivative-rule-sum","seed":202610085025,"semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"24\"],\"point_irrelevant\"],\"operation\":\"derivative_sum\"}"},"fields":{"title":"Derivative of a Polynomial Sum","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [-22, -1, 1, -1], starting with its constant term. Find the derivative of f+g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"24","explanation":"Use f′+g′. The exact result is 24; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Sum"],"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-20261007-advance-20261008-r05-derivative-rules-0026","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2942a7d4c978a31fab5cc259fcb7fb0183730710d805d90c89328da48fc462ff","family_id":"qf-derivative-rule-scalar","seed":202610085026,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[\"zero_derivative\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [1, 1], starting with its constant term. Find the derivative of (0)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use c*f′, including c=0. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0027","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541f16993df16051f596db4bc14e7908159c439c934debe1163dfccdb5d68b53","family_id":"qf-derivative-rule-scalar","seed":202610085027,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"2\",\"0\",\"2\"],\"-2\",\"-1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-16","explanation":"Use c*f′, including c=0. The exact result is -16; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0028","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f369695ad137cf4ed5ed5b3234007a94c4c8c5a9649c04eed826c46acda7c8a0","family_id":"qf-derivative-rule-scalar","seed":202610085028,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"3\",\"1\",\"3\"],\"3\",\"0\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"9","explanation":"Use c*f′, including c=0. The exact result is 9; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0029","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff025ffebacef1b63efa7925b07bdca6633ba66487795025191a817c4414f105","family_id":"qf-derivative-rule-scalar","seed":202610085029,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"4\",\"-1\",\"4\"],\"-4\",\"1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-56","explanation":"Use c*f′, including c=0. The exact result is -56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0030","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6daa86f5dbdda511db39ef05f9df9fee9c9923df92f9e98310cdd136489762b","family_id":"qf-derivative-rule-scalar","seed":202610085030,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"5\",\"0\",\"1\"],\"1\",\"2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"17","explanation":"Use c*f′, including c=0. The exact result is 17; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0031","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cddd8c3d320aca31245e0e9cdfdae07b52e2d0c5352a4cc705cea0bc6d924774","family_id":"qf-derivative-rule-scalar","seed":202610085031,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"6\",\"1\",\"2\"],\"-2\",\"-2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-52","explanation":"Use c*f′, including c=0. The exact result is -52; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0032","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"178adaf1672040c3e035fa7504a730c94ccd7a3d3f16f76c95ecd655488c0228","family_id":"qf-derivative-rule-scalar","seed":202610085032,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"7\",\"-1\",\"3\"],\"3\",\"-1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Find the derivative of (3)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"54","explanation":"Use c*f′, including c=0. The exact result is 54; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0033","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf187e9a8c87cb610dd192a5dab779c16b5b83661d98173db2890ed9ad2e6641","family_id":"qf-derivative-rule-scalar","seed":202610085033,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"8\",\"0\",\"4\"],\"-4\",\"0\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-32","explanation":"Use c*f′, including c=0. The exact result is -32; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0034","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8903f0b2b3e62c48e1419a799c120107d7fdf7a2b731fcde7b6e086216f77240","family_id":"qf-derivative-rule-scalar","seed":202610085034,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"9\",\"1\",\"1\"],\"1\",\"1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Find the derivative of (1)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"14","explanation":"Use c*f′, including c=0. The exact result is 14; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0035","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2871ddf43a472ac34b0f90d86232f9ddd6e236c30d7704528d793c7caac1d6a8","family_id":"qf-derivative-rule-scalar","seed":202610085035,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"10\",\"-1\",\"2\"],\"-2\",\"2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-60","explanation":"Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0036","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07e6c467e7b1f39b17a6d0880c8f4552560f5cab572d69a4a687d873e279ca3a","family_id":"qf-derivative-rule-scalar","seed":202610085036,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"11\",\"0\",\"3\"],\"3\",\"-2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Find the derivative of (3)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"141","explanation":"Use c*f′, including c=0. The exact result is 141; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0037","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"138dc5743cd12c82e7d5c692c3f07f15ddae15ca0f5a3bfd9047d4e59776f387","family_id":"qf-derivative-rule-scalar","seed":202610085037,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"12\",\"1\",\"4\"],\"-4\",\"-1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-88","explanation":"Use c*f′, including c=0. The exact result is -88; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0038","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"606d556de87b4f7b485835b7e8b797af12dff42040659e6433930c222112d1b0","family_id":"qf-derivative-rule-scalar","seed":202610085038,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"13\",\"-1\",\"1\"],\"1\",\"0\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Find the derivative of (1)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"13","explanation":"Use c*f′, including c=0. The exact result is 13; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0039","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d88666a8e80448ece9898b41e1b6dcee10b3625f50eef325af584a943887c99e","family_id":"qf-derivative-rule-scalar","seed":202610085039,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"14\",\"0\",\"2\"],\"-2\",\"1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Find the derivative of (-2)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-40","explanation":"Use c*f′, including c=0. The exact result is -40; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0040","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78203ec0bec34af482e19a528620b95312be70809fa3dd91b8fe11c0987c5d90","family_id":"qf-derivative-rule-scalar","seed":202610085040,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"15\",\"1\",\"3\"],\"3\",\"2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Find the derivative of (3)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"165","explanation":"Use c*f′, including c=0. The exact result is 165; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0041","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e4c26f9fe892846b0e67890c1c3bde33aad8ffb2aabab29c61c0be521ed05aa","family_id":"qf-derivative-rule-scalar","seed":202610085041,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"16\",\"-1\",\"4\"],\"-4\",\"-2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Find the derivative of (-4)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-272","explanation":"Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0042","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96c8b33bb2fb12dc0cb521e1be8e430c3c474c71acb467aed3730ab454fe5f47","family_id":"qf-derivative-rule-scalar","seed":202610085042,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"17\",\"0\",\"1\"],\"1\",\"-1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Find the derivative of (1)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"20","explanation":"Use c*f′, including c=0. The exact result is 20; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0043","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d80fa7af04921b411fa9f99c882fe66f78557d46993ea29a54f64e002c85f81","family_id":"qf-derivative-rule-scalar","seed":202610085043,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"18\",\"1\",\"2\"],\"-2\",\"0\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Find the derivative of (-2)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-36","explanation":"Use c*f′, including c=0. The exact result is -36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0044","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"221c8844f4f5d3eba7f29c5a0524ae5882ff87c2470a42d0db57132a1f6f9e81","family_id":"qf-derivative-rule-scalar","seed":202610085044,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"19\",\"-1\",\"3\"],\"3\",\"1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Find the derivative of (3)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"78","explanation":"Use c*f′, including c=0. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0045","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb3997d7f3d8102109d6a87c2a7c8e76cd0a7df3fe088c22e517823ed68f0ed6","family_id":"qf-derivative-rule-scalar","seed":202610085045,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"20\",\"0\",\"4\"],\"-4\",\"2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Find the derivative of (-4)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-272","explanation":"Use c*f′, including c=0. The exact result is -272; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0046","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3ade24e5271c505f0730d5cf975d61bfa629b8da3dfec33101dc2d623b148b5","family_id":"qf-derivative-rule-scalar","seed":202610085046,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"21\",\"1\",\"1\"],\"1\",\"-2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Find the derivative of (1)f at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"29","explanation":"Use c*f′, including c=0. The exact result is 29; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0047","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00d3ee04c662b9b8e9f04d6e1eadfbe487ce557284ba18bb6914daec983f7bd2","family_id":"qf-derivative-rule-scalar","seed":202610085047,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"22\",\"-1\",\"2\"],\"-2\",\"-1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Find the derivative of (-2)f at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-60","explanation":"Use c*f′, including c=0. The exact result is -60; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0048","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"866845b8e53fed8f3b22a7bcdd4ec476f6ec22ee9a725dfb3f1df230deaadceb","family_id":"qf-derivative-rule-scalar","seed":202610085048,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"23\",\"0\",\"3\"],\"3\",\"0\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Find the derivative of (3)f at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"69","explanation":"Use c*f′, including c=0. The exact result is 69; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0049","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a816ca6f12cc9107c3c94fbe40d95a6c37d794a1fe2ded0b0988935535c70524","family_id":"qf-derivative-rule-scalar","seed":202610085049,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"24\",\"1\",\"4\"],\"-4\",\"1\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Find the derivative of (-4)f at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-152","explanation":"Use c*f′, including c=0. The exact result is -152; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0050","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"676a6a2714475660682f096ec68fc90d6164db47189eb212cf99f6dbffa6d99e","family_id":"qf-derivative-rule-scalar","seed":202610085050,"semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"25\",\"-1\",\"1\"],\"1\",\"2\"],\"operation\":\"derivative_scalar\"}"},"fields":{"title":"Derivative of a Scalar Multiple of a Polynomial","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Find the derivative of (1)f at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"33","explanation":"Use c*f′, including c=0. The exact result is 33; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Scalar Multiple of a Polynomial"],"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-20261007-advance-20261008-r05-derivative-rules-0051","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad009e6e992f40037a0c110955d3e5c1bfd4c4aabea116269a71470335174a2","family_id":"qf-derivative-rule-product","seed":202610085051,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[\"constant_product\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [3], starting with its constant term; g has ascending coefficients [-2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use f′g+fg′, not f′g′. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0052","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"124b4a02ace262fdf303616c5bbc3dd607ca1e2252cd299e7d5488e429058a69","family_id":"qf-derivative-rule-product","seed":202610085052,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-11\",\"2\",\"0\",\"2\"],[\"2\",\"0\",\"1\",\"3\"]],\"-1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-105","explanation":"Use f′g+fg′, not f′g′. The exact result is -105; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0053","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c1eb1df0d42b65fa3e4958d31d892f279e069e7959520c8fa804e0489c6d6c2","family_id":"qf-derivative-rule-product","seed":202610085053,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-10\",\"3\",\"1\",\"3\"],[\"2\",\"1\",\"1\",\"4\"]],\"0\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-4","explanation":"Use f′g+fg′, not f′g′. The exact result is -4; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0054","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54641e5f5c58321ecc67437f2e04be0b00c8dc32273a8145e06ac1370f0a2a2f","family_id":"qf-derivative-rule-product","seed":202610085054,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-9\",\"4\",\"-1\",\"4\"],[\"2\",\"2\",\"1\",\"2\"]],\"1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"78","explanation":"Use f′g+fg′, not f′g′. The exact result is 78; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0055","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0df0935dc000caa0fd8886e76d082568e8a507e020d457ddad2c3f6220afa164","family_id":"qf-derivative-rule-product","seed":202610085055,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-8\",\"5\",\"0\",\"1\"],[\"2\",\"-1\",\"1\",\"3\"]],\"2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"866","explanation":"Use f′g+fg′, not f′g′. The exact result is 866; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0056","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dcd1fabb67528180fe4ddffa2e022c98314d76aef8d4eb54477886fcc7c097c","family_id":"qf-derivative-rule-product","seed":202610085056,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-7\",\"6\",\"1\",\"2\"],[\"2\",\"0\",\"1\",\"4\"]],\"-2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-2040","explanation":"Use f′g+fg′, not f′g′. The exact result is -2040; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0057","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d48a92abe3893001b1e5b20447ee60d008c0f51b4cc3fc8b8ee5222d4e976f60","family_id":"qf-derivative-rule-product","seed":202610085057,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-6\",\"7\",\"-1\",\"3\"],[\"2\",\"1\",\"1\",\"2\"]],\"-1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-85","explanation":"Use f′g+fg′, not f′g′. The exact result is -85; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0058","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"116bd80b5850bb48612dd3699b701f82cef7096a58139d48101e14126bd0cd5d","family_id":"qf-derivative-rule-product","seed":202610085058,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-5\",\"8\",\"0\",\"4\"],[\"2\",\"2\",\"1\",\"3\"]],\"0\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"6","explanation":"Use f′g+fg′, not f′g′. The exact result is 6; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0059","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf7cc442674d246c65293eba742c39ff5753ab4a1d0394f16404bb8652ad65b","family_id":"qf-derivative-rule-product","seed":202610085059,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-4\",\"9\",\"1\",\"1\"],[\"2\",\"-1\",\"1\",\"4\"]],\"1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"175","explanation":"Use f′g+fg′, not f′g′. The exact result is 175; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0060","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b4690d3f9e8a4124876e32a32b713d28b9a16fa9eb5a2b3eb47a2059b2df864","family_id":"qf-derivative-rule-product","seed":202610085060,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-3\",\"10\",\"-1\",\"2\"],[\"2\",\"0\",\"1\",\"2\"]],\"2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"1472","explanation":"Use f′g+fg′, not f′g′. The exact result is 1472; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0061","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a0b962077cb440bfd0d42fc6c84fb5f99393f2ef379ec8506c12b90959e8e59","family_id":"qf-derivative-rule-product","seed":202610085061,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-2\",\"11\",\"0\",\"3\"],[\"2\",\"1\",\"1\",\"3\"]],\"-2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-2524","explanation":"Use f′g+fg′, not f′g′. The exact result is -2524; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0062","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3ba7d953b8082eee125ef93eaa885f2bd2edb27de308835b8a625f3c7bd0ab","family_id":"qf-derivative-rule-product","seed":202610085062,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-1\",\"12\",\"1\",\"4\"],[\"2\",\"2\",\"1\",\"4\"]],\"-1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-258","explanation":"Use f′g+fg′, not f′g′. The exact result is -258; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0063","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46d891a32427d22000ae257cdca78920a55ac7dd35d026133a510319dbd53f92","family_id":"qf-derivative-rule-product","seed":202610085063,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"0\",\"13\",\"-1\",\"1\"],[\"2\",\"-1\",\"1\",\"2\"]],\"0\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"26","explanation":"Use f′g+fg′, not f′g′. The exact result is 26; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0064","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42fe75d133bd107c8e8e21b1691cba6fe1b1b35127637feff00ab2cf4400eb39","family_id":"qf-derivative-rule-product","seed":202610085064,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"1\",\"14\",\"0\",\"2\"],[\"2\",\"0\",\"1\",\"3\"]],\"1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"307","explanation":"Use f′g+fg′, not f′g′. The exact result is 307; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0065","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de5d9a72b4777c8de3c60d9ef743096be5dd361120a970c60e3fb103a5af2d83","family_id":"qf-derivative-rule-product","seed":202610085065,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"1\",\"1\",\"4\"],[\"2\",\"15\",\"1\",\"3\"]],\"2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 4], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"5380","explanation":"Use f′g+fg′, not f′g′. The exact result is 5380; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0066","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68c6ad41ac18b0860868ff0dcad96b8bf9a3cdc41a9f55cebec8e689ae23d9cc","family_id":"qf-derivative-rule-product","seed":202610085066,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"2\",\"1\",\"2\"],[\"3\",\"16\",\"-1\",\"4\"]],\"-2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 2], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-2382","explanation":"Use f′g+fg′, not f′g′. The exact result is -2382; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0067","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd379f7a1319c742dc56d16c38aa7166dbdc548049c0de1f2ab9ed9ddbe4c70c","family_id":"qf-derivative-rule-product","seed":202610085067,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"-1\",\"1\",\"3\"],[\"4\",\"17\",\"0\",\"1\"]],\"-1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 3], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-64","explanation":"Use f′g+fg′, not f′g′. The exact result is -64; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0068","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a64b51265e2bf59e2f7becbdb6a7849d02f128f6fddb58ff2852fd39d6f66637","family_id":"qf-derivative-rule-product","seed":202610085068,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"0\",\"1\",\"4\"],[\"5\",\"18\",\"1\",\"2\"]],\"0\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 4], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"36","explanation":"Use f′g+fg′, not f′g′. The exact result is 36; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0069","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f424837c09201b19aeabe0103e0c35261d4583410fb30e078c39fc98f1e0ca8a","family_id":"qf-derivative-rule-product","seed":202610085069,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"1\",\"1\",\"2\"],[\"6\",\"19\",\"-1\",\"3\"]],\"1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"399","explanation":"Use f′g+fg′, not f′g′. The exact result is 399; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0070","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf28e1ff100eebd1d027c4525ed13da9716e72ddb8e6fb37de28fd49d961483a","family_id":"qf-derivative-rule-product","seed":202610085070,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"2\",\"1\",\"3\"],[\"7\",\"20\",\"0\",\"4\"]],\"2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 3], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"5630","explanation":"Use f′g+fg′, not f′g′. The exact result is 5630; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0071","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b1b93f1bde8a62e505da1700e88019ed8759db8605c5686795639e3652c6a9","family_id":"qf-derivative-rule-product","seed":202610085071,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"-1\",\"1\",\"4\"],[\"8\",\"21\",\"1\",\"1\"]],\"-2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 4], starting with its constant term. Find the derivative of fg at x=-2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-2330","explanation":"Use f′g+fg′, not f′g′. The exact result is -2330; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0072","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eebd91d4a5a9fe0b0bd9a0350dc62f69ca60e6b34178a38eaab447a7c919d53e","family_id":"qf-derivative-rule-product","seed":202610085072,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"0\",\"1\",\"2\"],[\"9\",\"22\",\"-1\",\"2\"]],\"-1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 2], starting with its constant term. Find the derivative of fg at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-34","explanation":"Use f′g+fg′, not f′g′. The exact result is -34; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0073","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cc7827516e2e787bbfbfd4880c51dc968ad8427d9740c65858d133d71e7a059","family_id":"qf-derivative-rule-product","seed":202610085073,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"10\",\"23\",\"0\",\"3\"],[\"2\",\"1\",\"1\",\"3\"]],\"0\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term; g has ascending coefficients [2, 1, 1, 3], starting with its constant term. Find the derivative of fg at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"56","explanation":"Use f′g+fg′, not f′g′. The exact result is 56; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0074","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c9d6aba0b5d0c711c0a7c9fcc6f35629dbaedfab5e32c928d94edbc14306cdc","family_id":"qf-derivative-rule-product","seed":202610085074,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"11\",\"24\",\"1\",\"4\"],[\"2\",\"2\",\"1\",\"4\"]],\"1\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term; g has ascending coefficients [2, 2, 1, 4], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"982","explanation":"Use f′g+fg′, not f′g′. The exact result is 982; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0075","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6e7ca7b5f07feb6006e4ea944d9880cb19c53821e1a8f455c646b9848b5d652","family_id":"qf-derivative-rule-product","seed":202610085075,"semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"12\",\"25\",\"-1\",\"1\"],[\"2\",\"-1\",\"1\",\"2\"]],\"2\"],\"operation\":\"derivative_product\"}"},"fields":{"title":"Derivative of a Polynomial Product","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term; g has ascending coefficients [2, -1, 1, 2], starting with its constant term. Find the derivative of fg at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"2442","explanation":"Use f′g+fg′, not f′g′. The exact result is 2442; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Derivative of a Polynomial Product"],"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-20261007-advance-20261008-r05-derivative-rules-0076","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abb17673627bbe244073b24f22383d4b60dc622082ca04ba98553d9b585faf87","family_id":"qf-derivative-rule-chain","seed":202610085076,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[\"constant_composite\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [1, 2, 3], starting with its constant term; g has ascending coefficients [4], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0077","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7314f7ba5ddde76123dfbc3f3dbfdbaafca6900c811d66233270aa3305fcbc6c","family_id":"qf-derivative-rule-chain","seed":202610085077,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"1\",\"1\"],[\"-4\",\"1\",\"2\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-11, 2, 1, 1], starting with its constant term; g has ascending coefficients [-4, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"42","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 42; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0078","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c34570ba8989a59ac3072b4011240ba766f1aeba048b5444e57af555a9d95e22","family_id":"qf-derivative-rule-chain","seed":202610085078,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"2\",\"1\"],[\"-3\",\"1\",\"3\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-10, 3, 2, 1], starting with its constant term; g has ascending coefficients [-3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"230","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 230; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0079","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99119cd8998a27deadbcbc216198cd34ed1a64ae669bff6521de0cbb938edc9e","family_id":"qf-derivative-rule-chain","seed":202610085079,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"0\",\"1\"],[\"-2\",\"1\",\"1\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-9, 4, 0, 1], starting with its constant term; g has ascending coefficients [-2, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"7412","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 7412; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0080","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe033d43dcad79959b45f17c6b4ffab8edc29e1812d15f56b3b5416e07d9434","family_id":"qf-derivative-rule-chain","seed":202610085080,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"1\",\"1\"],[\"-1\",\"1\",\"2\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-8, 1, 1, 1], starting with its constant term; g has ascending coefficients [-1, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0081","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fe1e59c09fcd327e78a9f7bd63fc61ea4081c8529bca29bb6d78825ed49524b","family_id":"qf-derivative-rule-chain","seed":202610085081,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"2\",\"1\"],[\"0\",\"1\",\"3\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-7, 2, 2, 1], starting with its constant term; g has ascending coefficients [0, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"2","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0082","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2d6deff71578efa18dba2095921fe79cceec91b5ece2458699b4b772109585d","family_id":"qf-derivative-rule-chain","seed":202610085082,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"0\",\"1\"],[\"1\",\"1\",\"1\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-6, 3, 0, 1], starting with its constant term; g has ascending coefficients [1, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"306","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 306; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0083","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"006a3f5ce045899d8d439be035f20adf16d5691fe6ad6781043b649c5794958c","family_id":"qf-derivative-rule-chain","seed":202610085083,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"1\",\"1\"],[\"2\",\"1\",\"2\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-5, 4, 1, 1], starting with its constant term; g has ascending coefficients [2, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"26124","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 26124; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0084","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53e8c3d1b83bbf8f6b8e96a8a17c7c5ff848a1c84a64493ef302f09501ff3459","family_id":"qf-derivative-rule-chain","seed":202610085084,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"2\",\"1\"],[\"3\",\"1\",\"3\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-4, 1, 2, 1], starting with its constant term; g has ascending coefficients [3, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-130","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -130; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0085","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ebdf0358677d694b8ea97c6d70a125653db0b15e29341c2dc509d076acae13a","family_id":"qf-derivative-rule-chain","seed":202610085085,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"0\",\"1\"],[\"4\",\"1\",\"1\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-3, 2, 0, 1], starting with its constant term; g has ascending coefficients [4, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"50","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 50; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0086","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8a0c7cb6ce70f687a215865e245f962397f41a6437c50a06f2272e1942f46c","family_id":"qf-derivative-rule-chain","seed":202610085086,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"1\",\"1\"],[\"5\",\"1\",\"2\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-2, 3, 1, 1], starting with its constant term; g has ascending coefficients [5, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"2112","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 2112; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0087","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aeb65b1a36ba1adfdedd38b41c4fbb0f95e0ffb086b4c8d0c642d263d60e2ca","family_id":"qf-derivative-rule-chain","seed":202610085087,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"2\",\"1\"],[\"6\",\"1\",\"3\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [-1, 4, 2, 1], starting with its constant term; g has ascending coefficients [6, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"61700","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 61700; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0088","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e3f1914216bf4af7b5f63476c22fe1ee439cd6d60b502406b4d50fe4b1dc483","family_id":"qf-derivative-rule-chain","seed":202610085088,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"0\",\"1\"],[\"7\",\"1\",\"1\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [0, 1, 0, 1], starting with its constant term; g has ascending coefficients [7, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"218","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 218; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0089","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"914a2a15f17b5c27dd81cf304bcc4776692431d1a12d4fcd126749772ca453e6","family_id":"qf-derivative-rule-chain","seed":202610085089,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"1\",\"1\"],[\"8\",\"1\",\"2\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [1, 2, 1, 1], starting with its constant term; g has ascending coefficients [8, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"210","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 210; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0090","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37034107812296bb483c01787dda39444bd1f94d29131a66191af7518352ad0a","family_id":"qf-derivative-rule-chain","seed":202610085090,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"2\",\"1\"],[\"9\",\"1\",\"3\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [2, 3, 2, 1], starting with its constant term; g has ascending coefficients [9, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"6470","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 6470; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0091","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bc39b8202d0a5ebd4085f7de05fe2c98b7ed9be9fac5444a62a41ee1639878e","family_id":"qf-derivative-rule-chain","seed":202610085091,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"0\",\"1\"],[\"10\",\"1\",\"1\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [3, 4, 0, 1], starting with its constant term; g has ascending coefficients [10, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"29444","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 29444; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0092","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64b68267148ff104b414aec49f964423f82eade28f353f2ca3f2dd804eb452d","family_id":"qf-derivative-rule-chain","seed":202610085092,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"1\",\"1\"],[\"11\",\"1\",\"2\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [4, 1, 1, 1], starting with its constant term; g has ascending coefficients [11, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"0","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 0; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0093","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"208469a4362804f4764258a3c2f1f6832f7531b1e7d74d27ffe2d2901a391c25","family_id":"qf-derivative-rule-chain","seed":202610085093,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"2\",\"1\"],[\"12\",\"1\",\"3\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [5, 2, 2, 1], starting with its constant term; g has ascending coefficients [12, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"482","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 482; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0094","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43d5667a82129acbb714491899ee5203e5ea5a95b5012d660d10d89ca0c7c302","family_id":"qf-derivative-rule-chain","seed":202610085094,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"0\",\"1\"],[\"13\",\"1\",\"1\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [6, 3, 0, 1], starting with its constant term; g has ascending coefficients [13, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"4626","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 4626; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0095","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"984cd5c4acf87525543ce6d87b7815d55c785f8238f06e82918b059652cf67fa","family_id":"qf-derivative-rule-chain","seed":202610085095,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"1\",\"1\"],[\"14\",\"1\",\"2\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [7, 4, 1, 1], starting with its constant term; g has ascending coefficients [14, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"65940","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 65940; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0096","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a42f71ca9fbd03275820af9e639b6c1273aaa90c5e003c8931aac406961521","family_id":"qf-derivative-rule-chain","seed":202610085096,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"2\",\"1\"],[\"15\",\"1\",\"3\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [8, 1, 2, 1], starting with its constant term; g has ascending coefficients [15, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"-1666","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is -1666; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0097","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e905b0c315ca3d408ffa9d15fbf230b7b8c99a5be8170b8da22568804ac9b228","family_id":"qf-derivative-rule-chain","seed":202610085097,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"0\",\"1\"],[\"16\",\"1\",\"1\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [9, 2, 0, 1], starting with its constant term; g has ascending coefficients [16, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=0. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"770","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 770; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0098","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d3b21453bf386d16709106f12940e2146f74875e8100866fbc9323cb1a7c9a3","family_id":"qf-derivative-rule-chain","seed":202610085098,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"1\",\"1\"],[\"17\",\"1\",\"2\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [10, 3, 1, 1], starting with its constant term; g has ascending coefficients [17, 1, 2, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"10944","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 10944; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0099","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f59d127b8f8bf7632c5f2bd9d8cc459f006f5d4b9559597dc9714e585b0d8ecf","family_id":"qf-derivative-rule-chain","seed":202610085099,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"2\",\"1\"],[\"18\",\"1\",\"3\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [11, 4, 2, 1], starting with its constant term; g has ascending coefficients [18, 1, 3, 1], starting with its constant term. Find the derivative of f∘g at x=2. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"124100","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 124100; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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-20261007-advance-20261008-r05-derivative-rules-0100","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d680f96abb0b8626b94114497eded3c344bed5b4214c803b646adc591fc16032","family_id":"qf-derivative-rule-chain","seed":202610085100,"semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"0\",\"1\"],[\"19\",\"1\",\"1\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"},"fields":{"title":"Chain-Rule Derivative of a Polynomial Composition","stem":"On all R, f has ascending coefficients [12, 1, 0, 1], starting with its constant term; g has ascending coefficients [19, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=-1. Input degrees are at most three; retain all higher terms produced by the operation.","answer":{"kind":"short_text","value":"1946","explanation":"Use f′(g(x))*g′(x), retaining both the inner evaluation and derivative. The exact result is 1946; acceptance expands the complete resulting polynomial before differentiation, with no truncation of product or composition terms. These polynomial instances do not establish differentiability of arbitrary sampled functions."},"skills":["Chain-Rule Derivative of a Polynomial Composition"],"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."}}