{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r04-derivative-rules-0001","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6669b46625ec87b01307a396ab8822ab1d31b78239e0e708cbc6fb1ac807a51b","family_id":"qf-density-derivative-second-value","seed":202610094001,"semantic_template":"qf-density-derivative-second-value/[\"zero_second_derivative\"]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Differentiate twice and evaluate exactly, obtaining 0/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0002","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57b7b06019519f2cd6ed1d9cca3184f0466593a39d645995fad5292d368b8a60","family_id":"qf-density-derivative-second-value","seed":202610094002,"semantic_template":"qf-density-derivative-second-value/[[2],-1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-11, 2, 2, 0], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"Differentiate twice and evaluate exactly, obtaining 4/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0003","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"004db9f6b073ca473dac00a7272af9ca4bf11e657eae22ef6f69a4ba4e8da4d3","family_id":"qf-density-derivative-second-value","seed":202610094003,"semantic_template":"qf-density-derivative-second-value/[3]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-10, 3, 3, 1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"6/1","explanation":"Differentiate twice and evaluate exactly, obtaining 6/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0004","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c029bedeffe0bd6e7cd162ee1b203a984e6ab8bf7d5dd8a8ae87987a7e5da18b","family_id":"qf-density-derivative-second-value","seed":202610094004,"semantic_template":"qf-density-derivative-second-value/[[4,-1],1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-9, 1, 4, -1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"Differentiate twice and evaluate exactly, obtaining 2/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0005","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f2b60e2433a12fa3ff319e8823f4041a014de5e7f7118fbbf8decdfb6c7b95f","family_id":"qf-density-derivative-second-value","seed":202610094005,"semantic_template":"qf-density-derivative-second-value/[[5],2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-8, 2, 5, 0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"10/1","explanation":"Differentiate twice and evaluate exactly, obtaining 10/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0006","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c6fc36da0fdb5530e8bd5bca200f17f75b18108cc9770f6264f1f99162a5d49","family_id":"qf-density-derivative-second-value","seed":202610094006,"semantic_template":"qf-density-derivative-second-value/[[6,1],-2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-7, 3, 6, 1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Differentiate twice and evaluate exactly, obtaining 0/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0007","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a070a166441c9b20d16fc468c7d37945709d9cca5ddacba59bfb939abd542234","family_id":"qf-density-derivative-second-value","seed":202610094007,"semantic_template":"qf-density-derivative-second-value/[[7,-1],-1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-6, 1, 7, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"20/1","explanation":"Differentiate twice and evaluate exactly, obtaining 20/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0008","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e59444540b297a7d8fe2f4fd981e70ba13133cb34c1f182e3514fe42c130d557","family_id":"qf-density-derivative-second-value","seed":202610094008,"semantic_template":"qf-density-derivative-second-value/[8]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-5, 2, 8, 0], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"16/1","explanation":"Differentiate twice and evaluate exactly, obtaining 16/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0009","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e358dd57cdd09a6abe324526de3800b0934535f7ed7897a59493caab7584aa","family_id":"qf-density-derivative-second-value","seed":202610094009,"semantic_template":"qf-density-derivative-second-value/[[9,1],1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-4, 3, 9, 1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"24/1","explanation":"Differentiate twice and evaluate exactly, obtaining 24/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0010","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e360def6da26173f4aa15c884f5a21039f61c468d348a2206bf287b7e5dd7ebf","family_id":"qf-density-derivative-second-value","seed":202610094010,"semantic_template":"qf-density-derivative-second-value/[[10,-1],2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-3, 1, 10, -1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"8/1","explanation":"Differentiate twice and evaluate exactly, obtaining 8/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0011","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f714ddb7fdbae830370d806c3c59eef3695121f7fc89cdb8bf865c51ccf2ca7","family_id":"qf-density-derivative-second-value","seed":202610094011,"semantic_template":"qf-density-derivative-second-value/[[11],-2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-2, 2, 11, 0], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"22/1","explanation":"Differentiate twice and evaluate exactly, obtaining 22/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0012","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7b28a26c3e2a687feed2ea5f615bead8bcd2f3e752daf456ffabc521f8c962b","family_id":"qf-density-derivative-second-value","seed":202610094012,"semantic_template":"qf-density-derivative-second-value/[[12,1],-1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [-1, 3, 12, 1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"18/1","explanation":"Differentiate twice and evaluate exactly, obtaining 18/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0013","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"887262e57720045c9fe0f0faed6b9033212e83caf3e5e592b4825c1b475583c5","family_id":"qf-density-derivative-second-value","seed":202610094013,"semantic_template":"qf-density-derivative-second-value/[13]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [0, 1, 13, -1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"26/1","explanation":"Differentiate twice and evaluate exactly, obtaining 26/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0014","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4552dca396ba15331f6bf7610e486f91c7a605a5c7cb92ca66df34e26991e3b","family_id":"qf-density-derivative-second-value","seed":202610094014,"semantic_template":"qf-density-derivative-second-value/[[14],1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [1, 2, 14, 0], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"28/1","explanation":"Differentiate twice and evaluate exactly, obtaining 28/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0015","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb9e86188f84c2631fc2af5573936b5eccb4b460cd58947dc425e015b53be670","family_id":"qf-density-derivative-second-value","seed":202610094015,"semantic_template":"qf-density-derivative-second-value/[[15,1],2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [2, 3, 15, 1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"42/1","explanation":"Differentiate twice and evaluate exactly, obtaining 42/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0016","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1512c950bb29209be74792a59a03e0a13e3854df8a2427b1d85ba28ecc22442e","family_id":"qf-density-derivative-second-value","seed":202610094016,"semantic_template":"qf-density-derivative-second-value/[[16,-1],-2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [3, 1, 16, -1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"44/1","explanation":"Differentiate twice and evaluate exactly, obtaining 44/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0017","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29494ef21a3d0ac2e41bc163feb5c6ede461f238eaf1d7f8b6d6a09632447602","family_id":"qf-density-derivative-second-value","seed":202610094017,"semantic_template":"qf-density-derivative-second-value/[[17],-1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [4, 2, 17, 0], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"34/1","explanation":"Differentiate twice and evaluate exactly, obtaining 34/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0018","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f18afbf8a6abba5612a7972c85a6f747b31ca3c5161bdd0e57b99b3e9df2c19","family_id":"qf-density-derivative-second-value","seed":202610094018,"semantic_template":"qf-density-derivative-second-value/[18]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [5, 3, 18, 1], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"36/1","explanation":"Differentiate twice and evaluate exactly, obtaining 36/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0019","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66cab684112e96abc65f0f7c834595b0112aed3f2700461cbef6fe9c31e1897b","family_id":"qf-density-derivative-second-value","seed":202610094019,"semantic_template":"qf-density-derivative-second-value/[[19,-1],1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [6, 1, 19, -1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"32/1","explanation":"Differentiate twice and evaluate exactly, obtaining 32/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0020","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcd90968010e64f763bf5e68a28233183639387341b07bb1ba838805f15f2a1b","family_id":"qf-density-derivative-second-value","seed":202610094020,"semantic_template":"qf-density-derivative-second-value/[[20],2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [7, 2, 20, 0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"40/1","explanation":"Differentiate twice and evaluate exactly, obtaining 40/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0021","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f96ebbc46699ec1204b0ac6fdf4e91d550581facfc7d338e28a751618cabe19b","family_id":"qf-density-derivative-second-value","seed":202610094021,"semantic_template":"qf-density-derivative-second-value/[[21,1],-2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [8, 3, 21, 1], constant term first. Find f″(-2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"30/1","explanation":"Differentiate twice and evaluate exactly, obtaining 30/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0022","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40cce257122826465c3bb30d19eb422b78a1196e40586871f9f692ee965fb67","family_id":"qf-density-derivative-second-value","seed":202610094022,"semantic_template":"qf-density-derivative-second-value/[[22,-1],-1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [9, 1, 22, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"50/1","explanation":"Differentiate twice and evaluate exactly, obtaining 50/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0023","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b53c64b6b2876ac00a238c4d578d67864bc0b3cf0aecf16699b03eb2859c7a8","family_id":"qf-density-derivative-second-value","seed":202610094023,"semantic_template":"qf-density-derivative-second-value/[23]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [10, 2, 23, 0], constant term first. Find f″(0). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"46/1","explanation":"Differentiate twice and evaluate exactly, obtaining 46/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0024","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a5c829397d1e0ffd406a5ba70ff143c4531d42e2310da728a9e79b0fd2bbd6b","family_id":"qf-density-derivative-second-value","seed":202610094024,"semantic_template":"qf-density-derivative-second-value/[[24,1],1]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [11, 3, 24, 1], constant term first. Find f″(1). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"54/1","explanation":"Differentiate twice and evaluate exactly, obtaining 54/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0025","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10d5c0929f6abf1cae6f73e75f8f12d2c3ae55bc6862d726d88d707bf468b2ad","family_id":"qf-density-derivative-second-value","seed":202610094025,"semantic_template":"qf-density-derivative-second-value/[[25,-1],2]"},"fields":{"title":"Second Derivative of a Polynomial at a Point","stem":"On all R, f has ascending coefficients [12, 1, 25, -1], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"38/1","explanation":"Differentiate twice and evaluate exactly, obtaining 38/1. Constant and linear inputs have zero second derivative; this declared polynomial calculation is not sampling of arbitrary functions."},"skills":["Second Derivative of a Polynomial at a Point"],"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-20261009-r04-derivative-rules-0026","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daedf5d898a8f2197ddd4b6f6751e47ce52af61b7f9c7ad3af9007f038508760","family_id":"qf-density-derivative-product-candidate","seed":202610094026,"semantic_template":"qf-density-derivative-product-candidate/[\"zero_derivative\",0]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [0], constant term first and g has ascending coefficients [2, 1], constant term first. At x=3, is 0 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0027","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2894081a942b26e3277c08295c5ba85e87101a248db5981026b51377b360baa4","family_id":"qf-density-derivative-product-candidate","seed":202610094027,"semantic_template":"qf-density-derivative-product-candidate/[\"zero_derivative\",1]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [5], constant term first and g has ascending coefficients [-2], constant term first. At x=1, is 1 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0028","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c3110578c8b4be38a329b0f747c109e12b9fe927d701a6803e066c551c2ef91","family_id":"qf-density-derivative-product-candidate","seed":202610094028,"semantic_template":"qf-density-derivative-product-candidate/[[[-10,3,1,3],[2,-1,1,3]],1,40]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=1, is 40 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0029","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4be84347178eb894750647667be4d4c3792d8e25887116dae6c05c3e95ef56e","family_id":"qf-density-derivative-product-candidate","seed":202610094029,"semantic_template":"qf-density-derivative-product-candidate/[[[-9,4,-1,4],[2,-1,1,1]],2,981]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 981 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0030","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf14cc69f2ca41f2a3464a281bb7ef7110425f56ab5b75a1d754b383f9ec9cf3","family_id":"qf-density-derivative-product-candidate","seed":202610094030,"semantic_template":"qf-density-derivative-product-candidate/[[[-8,5,0,1],[2,-1,1,2]],-1,-25]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -25 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0031","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce14b0edadff561bef2440f37fdc9a8fb33a541a9edbeef6273e1b624498c7d4","family_id":"qf-density-derivative-product-candidate","seed":202610094031,"semantic_template":"qf-density-derivative-product-candidate/[[[-7,6,1,2],[2,-1,1,3]],0,19]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 19 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0032","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8912761d9c214f41bc1122e8e4d7a31e775f72b0d1ee47d63a407457853e8ec9","family_id":"qf-density-derivative-product-candidate","seed":202610094032,"semantic_template":"qf-density-derivative-product-candidate/[[[-6,7,-1,3],[2,-1,1,1]],1,54]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=1, is 54 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0033","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2fdab8345d197447b85167d5c7b53a84f7b1ba51f3552bc94b575dcdd7aec77","family_id":"qf-density-derivative-product-candidate","seed":202610094033,"semantic_template":"qf-density-derivative-product-candidate/[[[-5,8,0,4],[2,-1,1,2]],2,2282]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=2, is 2282 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0034","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0e9a4589f7ffb5c48b749253f52c82e99f33f0c95037c287005d8f0561923c","family_id":"qf-density-derivative-product-candidate","seed":202610094034,"semantic_template":"qf-density-derivative-product-candidate/[[[-4,9,1,1],[2,-1,1,3]],-1,-68]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=-1, is -68 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0035","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1fe2cb6cdbe79f5dd904a85add6408d193817d2f10dabce1c25d67bdc20ca0c","family_id":"qf-density-derivative-product-candidate","seed":202610094035,"semantic_template":"qf-density-derivative-product-candidate/[[[-3,10,-1,2],[2,-1,1,1]],0,23]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=0, is 23 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0036","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cb0eb87dd92d45f6a60297c6549a1b087f5098434d05e69a9b4a5de1ab2a08a","family_id":"qf-density-derivative-product-candidate","seed":202610094036,"semantic_template":"qf-density-derivative-product-candidate/[[[-2,11,0,3],[2,-1,1,2]],1,165]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=1, is 165 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0037","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830055c57626317a2b0e9ce2e4cd447584a7a8afc30c762036e488760479a614","family_id":"qf-density-derivative-product-candidate","seed":202610094037,"semantic_template":"qf-density-derivative-product-candidate/[[[-1,12,1,4],[2,-1,1,3]],2,4093]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=2, is 4093 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0038","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353a4c5b23c4dce5bb5732cb84ee5570207e33910781dc31ab3493cf9af112ed","family_id":"qf-density-derivative-product-candidate","seed":202610094038,"semantic_template":"qf-density-derivative-product-candidate/[[[0,13,-1,1],[2,-1,1,1]],-1,54]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=-1, is 54 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0039","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ebf61476f3b18a1e7fc283a37e6e59ce94811ec3bf282b4b719337c25bf562f","family_id":"qf-density-derivative-product-candidate","seed":202610094039,"semantic_template":"qf-density-derivative-product-candidate/[[[1,14,0,2],[2,-1,1,2]],0,28]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=0, is 28 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0040","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"741625a966f20db40b29b60d41c1b84c9e3c217359909762fb0f791e92199761","family_id":"qf-density-derivative-product-candidate","seed":202610094040,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[2,15,1,3]],1,340]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=1, is 340 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0041","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69aea847bbf50fdfd0cfa2d7aeeccff0bf17db64592768d8dd008e5e12be2ea1","family_id":"qf-density-derivative-product-candidate","seed":202610094041,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[3,16,-1,4]],2,1665]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 1665 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0042","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fb8af456ebfafa238d0dfbb3671a5617f93735ae0e946cb60a5ee30a37aaea2","family_id":"qf-density-derivative-product-candidate","seed":202610094042,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,2],[4,17,0,1]],-1,-1]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -1 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0043","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b5574e5f6ca5bfd7890e1913a8f4aa9fd12229e873bf3d3261788e662fce8dc","family_id":"qf-density-derivative-product-candidate","seed":202610094043,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[5,18,1,2]],0,31]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 31 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0044","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e61ad730d909f68b3c4284bce478fd236c3934df09d38e5a8420ec8c7612c1b","family_id":"qf-density-derivative-product-candidate","seed":202610094044,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[6,19,-1,3]],1,186]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=1, is 186 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0045","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e30f2385369cef98f049c4e5c35ee557220a6409a43aba578a10f0c0b04daca7","family_id":"qf-density-derivative-product-candidate","seed":202610094045,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,2],[7,20,0,4]],2,3494]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=2, is 3494 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0046","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aefec76b924327f5f0abdb165c983456212fc646a196b9d4feb6c6aed0900edd","family_id":"qf-density-derivative-product-candidate","seed":202610094046,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[8,21,1,1]],-1,-56]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=-1, is -56 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0047","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"462e07268528f708597647f1101b34a3128b0255cf863d135ea7c74cbd3f6c42","family_id":"qf-density-derivative-product-candidate","seed":202610094047,"semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[9,22,-1,2]],0,35]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=0, is 35 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0048","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec6d2bbac1bbbc6058040ed2a1dae941817bf1dabbe4cbdfd5e0a125212c4ba6","family_id":"qf-density-derivative-product-candidate","seed":202610094048,"semantic_template":"qf-density-derivative-product-candidate/[[[10,23,0,3],[2,-1,1,2]],1,381]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=1, is 381 the derivative of fg?","answer":{"kind":"short_text","value":"No","explanation":"Compare the candidate with the exact derivative of the complete product, giving No. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0049","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60eb91ed312daa2bb876ff1f86b0987ca1787b85cf135d9c066fdcea8d5ba1dc","family_id":"qf-density-derivative-product-candidate","seed":202610094049,"semantic_template":"qf-density-derivative-product-candidate/[[[11,24,1,4],[2,-1,1,3]],2,5833]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=2, is 5833 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0050","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ecaf55ed400bed9427d6ebf293c064588df12c5ee006275a837bfd819bc598b","family_id":"qf-density-derivative-product-candidate","seed":202610094050,"semantic_template":"qf-density-derivative-product-candidate/[[[12,25,-1,1],[2,-1,1,1]],-1,90]"},"fields":{"title":"Eligibility of a Candidate Product-Derivative Value","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=-1, is 90 the derivative of fg?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."},"skills":["Eligibility of a Candidate Product-Derivative Value"],"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-20261009-r04-derivative-rules-0051","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33a73407c3cd04826d750b831dcdf74a15fe0e4b8b1358fee268db4575aa7f09","family_id":"qf-density-derivative-chain-order-pair","seed":202610094051,"semantic_template":"qf-density-derivative-chain-order-pair/[\"both_constant_compositions\"]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [3], constant term first and g has ascending coefficients [1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (0/1, 0/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0052","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f820e848d4817cedd39b512a358b6f4dede0f8dbfd36ecaae25f040a65ce8f1","family_id":"qf-density-derivative-chain-order-pair","seed":202610094052,"semantic_template":"qf-density-derivative-chain-order-pair/[[-11,2,0,2],[-4,1,2,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-11, 2, 0, 2], constant term first and g has ascending coefficients [-4, 1, 2, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(98/1, 640/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (98/1, 640/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0053","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2908514c2806d65db65d72ccb45c4892444d0f964a89ba050dcd6c1a689395d1","family_id":"qf-density-derivative-chain-order-pair","seed":202610094053,"semantic_template":"qf-density-derivative-chain-order-pair/[[-10,3,1,3],[-3,1,3,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first and g has ascending coefficients [-3, 1, 3, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(430/1, 140/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (430/1, 140/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0054","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"163040410ef3331cf3f8292ee1e4fe92a24a045817e62011ed6c45c6a3fedde3","family_id":"qf-density-derivative-chain-order-pair","seed":202610094054,"semantic_template":"qf-density-derivative-chain-order-pair/[[-9,4,-1,4],[-2,1,1,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first and g has ascending coefficients [-2, 1, 1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(29036/1, 107616/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (29036/1, 107616/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0055","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f2311264215be92b03a8563aa27be66cfbc7c5d52879387d71b499667ccbbae","family_id":"qf-density-derivative-chain-order-pair","seed":202610094055,"semantic_template":"qf-density-derivative-chain-order-pair/[[-8,5,0,1],[-1,1,2,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [-1, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 4264/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (0/1, 4264/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0056","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88372c3e0f24e33b12d19a553288a5a99f6ce7a912a9eedfd1e35794e2dd3dea","family_id":"qf-density-derivative-chain-order-pair","seed":202610094056,"semantic_template":"qf-density-derivative-chain-order-pair/[[-7,6,1,2],[0,1,3,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [0, 1, 3, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/1, 636/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (6/1, 636/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0057","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32387759721dc7c3a55b7094c8ec64135dc2afe778dd004f21a7db1d0187435b","family_id":"qf-density-derivative-chain-order-pair","seed":202610094057,"semantic_template":"qf-density-derivative-chain-order-pair/[[-6,7,-1,3],[1,1,1,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first and g has ascending coefficients [1, 1, 1, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(858/1, 476/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (858/1, 476/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0058","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55e2a8c8b4658670926364b9852fbe0127256493acf54d9cbdfc9690ba24fbfb","family_id":"qf-density-derivative-chain-order-pair","seed":202610094058,"semantic_template":"qf-density-derivative-chain-order-pair/[[-5,8,0,4],[2,1,2,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, 1, 2, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(100968/1, 320320/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (100968/1, 320320/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0059","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b60b7df925a5ac5558b0f44e8fe25ff2b6b9aaa5ec3b26f58c58a8ddeb299d8","family_id":"qf-density-derivative-chain-order-pair","seed":202610094059,"semantic_template":"qf-density-derivative-chain-order-pair/[[-4,9,1,1],[3,1,3,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first and g has ascending coefficients [3, 1, 3, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-130/1, 4300/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (-130/1, 4300/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0060","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bc0cbef26bd3239fe27377a002ac8a4126307518394299f978f64f4112441ae","family_id":"qf-density-derivative-chain-order-pair","seed":202610094060,"semantic_template":"qf-density-derivative-chain-order-pair/[[-3,10,-1,2],[4,1,1,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first and g has ascending coefficients [4, 1, 1, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(98/1, 220/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (98/1, 220/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0061","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6490badcfafc9415e6e7ab509019e52af169620a6a912bdf638ef78d3228683","family_id":"qf-density-derivative-chain-order-pair","seed":202610094061,"semantic_template":"qf-density-derivative-chain-order-pair/[[-2,11,0,3],[5,1,2,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first and g has ascending coefficients [5, 1, 2, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(5920/1, 9620/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (5920/1, 9620/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0062","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55d9bb5ef84885897a53c3bcaa68b83cc4485a9eb3918c92e66e9504cd4d006e","family_id":"qf-density-derivative-chain-order-pair","seed":202610094062,"semantic_template":"qf-density-derivative-chain-order-pair/[[-1,12,1,4],[6,1,3,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first and g has ascending coefficients [6, 1, 3, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(236900/1, 691072/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (236900/1, 691072/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0063","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b444cdef9ed112847b79849440150a9f7092b619353281f903d8b5004ab72e","family_id":"qf-density-derivative-chain-order-pair","seed":202610094063,"semantic_template":"qf-density-derivative-chain-order-pair/[[0,13,-1,1],[7,1,1,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], constant term first and g has ascending coefficients [7, 1, 1, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(218/1, 11628/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (218/1, 11628/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0064","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbd604d0a3937a4afcb401fef5cab351375167a3215934c90563c6dbf764a04","family_id":"qf-density-derivative-chain-order-pair","seed":202610094064,"semantic_template":"qf-density-derivative-chain-order-pair/[[1,14,0,2],[8,1,2,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], constant term first and g has ascending coefficients [8, 1, 2, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(398/1, 112/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (398/1, 112/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0065","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e46a8b732289d6eb7bd3dc4568d124caae154fddb613650e18036007235e3caa","family_id":"qf-density-derivative-chain-order-pair","seed":202610094065,"semantic_template":"qf-density-derivative-chain-order-pair/[[2,15,1,3],[9,1,3,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], constant term first and g has ascending coefficients [9, 1, 3, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(18070/1, 37700/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (18070/1, 37700/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0066","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e22c9a69f9f26ae93a8d74f846676dbd36a2469fffdf0d76d8e8c7e4a1e818c7","family_id":"qf-density-derivative-chain-order-pair","seed":202610094066,"semantic_template":"qf-density-derivative-chain-order-pair/[[3,16,-1,4],[10,1,1,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [10, 1, 1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(116960/1, 722040/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (116960/1, 722040/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0067","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b42353ac35f97bef1fc8fcfc05c04af2788f7204026d79211be7974a36aad3","family_id":"qf-density-derivative-chain-order-pair","seed":202610094067,"semantic_template":"qf-density-derivative-chain-order-pair/[[4,17,0,1],[11,1,2,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], constant term first and g has ascending coefficients [11, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 10660/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (0/1, 10660/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0068","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c339ed6930c292eaec737864d12b7cb336ee5b6693acdeadb49ba871ae82bca6","family_id":"qf-density-derivative-chain-order-pair","seed":202610094068,"semantic_template":"qf-density-derivative-chain-order-pair/[[5,18,1,2],[12,1,3,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], constant term first and g has ascending coefficients [12, 1, 3, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(906/1, 1908/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (906/1, 1908/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0069","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c639158a237e5d2339e2f787be975847b6e24da1d434b2784f79b5bde62fb06","family_id":"qf-density-derivative-chain-order-pair","seed":202610094069,"semantic_template":"qf-density-derivative-chain-order-pair/[[6,19,-1,3],[13,1,1,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], constant term first and g has ascending coefficients [13, 1, 1, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(13746/1, 58292/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (13746/1, 58292/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0070","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e78d76fe82c81698ced243b08e83d908575f7bb38abf5d9d5a64dfc6f2708b54","family_id":"qf-density-derivative-chain-order-pair","seed":202610094070,"semantic_template":"qf-density-derivative-chain-order-pair/[[7,20,0,4],[14,1,2,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], constant term first and g has ascending coefficients [14, 1, 2, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(258468/1, 1294720/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (258468/1, 1294720/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0071","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26a5d8d072ac1018eef52aa3493f4b0ea5f47a03a6125811f5172125d7eb09bd","family_id":"qf-density-derivative-chain-order-pair","seed":202610094071,"semantic_template":"qf-density-derivative-chain-order-pair/[[8,21,1,1],[15,1,3,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], constant term first and g has ascending coefficients [15, 1, 3, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1642/1, 9460/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (-1642/1, 9460/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0072","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0df75a5a402156ee02910838aaeb64d5671293e74bc74c01a9553202fcf6e464","family_id":"qf-density-derivative-chain-order-pair","seed":202610094072,"semantic_template":"qf-density-derivative-chain-order-pair/[[9,22,-1,2],[16,1,1,1],0]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], constant term first and g has ascending coefficients [16, 1, 1, 1], constant term first. At x=0, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1526/1, 5764/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (1526/1, 5764/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0073","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee19e350aa8e0665f5f5a5ba39b4fdba3a13496c05b1f639e1f0179ed548c7f0","family_id":"qf-density-derivative-chain-order-pair","seed":202610094073,"semantic_template":"qf-density-derivative-chain-order-pair/[[10,23,0,3],[17,1,2,1],1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], constant term first and g has ascending coefficients [17, 1, 2, 1], constant term first. At x=1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(31936/1, 129056/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (31936/1, 129056/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0074","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bed912e9b57e100480c4ea6bf66f53313e81b9e4f5f2d8648b386cf7c8dad41","family_id":"qf-density-derivative-chain-order-pair","seed":202610094074,"semantic_template":"qf-density-derivative-chain-order-pair/[[11,24,1,4],[18,1,3,1],2]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], constant term first and g has ascending coefficients [18, 1, 3, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(482600/1, 2101096/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (482600/1, 2101096/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0075","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84a00ff499ab44d0c1e13756c46507472942feeeb1b81bd0744f55f0093c1a53","family_id":"qf-density-derivative-chain-order-pair","seed":202610094075,"semantic_template":"qf-density-derivative-chain-order-pair/[[12,25,-1,1],[19,1,1,1],-1]"},"fields":{"title":"Derivatives in Both Composition Orders","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], constant term first and g has ascending coefficients [19, 1, 1, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1922/1, 19380/1)","explanation":"Expand and differentiate each ordered composition without truncation to obtain (1922/1, 19380/1). Outer derivatives are evaluated at the corresponding inner values; the two orders cannot generally be exchanged."},"skills":["Derivatives in Both Composition Orders"],"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-20261009-r04-derivative-rules-0076","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f114bf06ac9568bc91c59d5ae6839cdbdc8617b2bcdead2be68e8d8364aee1bb","family_id":"qf-density-derivative-sum-coefficients","seed":202610094076,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,-1],[0,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [2, 1], constant term first and g has ascending coefficients [-4, -1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[0]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [0]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0077","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"567d67be80e495e01ca585b3ea01140ea19ff6567bcd106a6fee376863b71a1c","family_id":"qf-density-derivative-sum-coefficients","seed":202610094077,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,1,3,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 1, 3, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[2, 8, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [2, 8, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0078","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b708a3b1c682cd4f62b38c8f5fa8db372869610916c5d236fecd5aac6f6efcd0","family_id":"qf-density-derivative-sum-coefficients","seed":202610094078,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,2,4,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 2, 4, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[3, 10, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [3, 10, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0079","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"060d7dfef9208e2b6d8bc00e6db33daec4f60e6ac82d756cec5ded6fea49794c","family_id":"qf-density-derivative-sum-coefficients","seed":202610094079,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,3,5,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 3, 5, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[4, 12, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [4, 12, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0080","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c4eaf4d23e3a1b6b44c1a30aeceaed9a9a8365a33a6ef2b982e6d44860a3e23","family_id":"qf-density-derivative-sum-coefficients","seed":202610094080,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,4,6,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 4, 6, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[5, 14, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [5, 14, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0081","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b90234377eae398abd1ca71eb679c11c6ccb806ce0c7d1360baf4263f7c724da","family_id":"qf-density-derivative-sum-coefficients","seed":202610094081,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,5,7,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 5, 7, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[6, 16, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [6, 16, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0082","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af196cad5f3b68b1ea4d5df9d4bfa4231c850c226278977b5939e12e714a9765","family_id":"qf-density-derivative-sum-coefficients","seed":202610094082,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,6,8,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 6, 8, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[7, 18, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [7, 18, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0083","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"290a215ef51ca80fe16f6ae72876c768be482b021e7ed8f570e31a89cd94fc91","family_id":"qf-density-derivative-sum-coefficients","seed":202610094083,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,7,9,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 7, 9, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[8, 20, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [8, 20, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0084","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cff2b2473aa159faa6440c5b16e7efc7eddad0c9a79d6d40c20797666d2af3e","family_id":"qf-density-derivative-sum-coefficients","seed":202610094084,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,8,10,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 8, 10, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[9, 22, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [9, 22, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0085","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffe23302f491eb2209be5840634495bbbdcc9f44c923ee9d2b9f57b6ddafe4b3","family_id":"qf-density-derivative-sum-coefficients","seed":202610094085,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,9,11,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 9, 11, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[10, 24, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [10, 24, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0086","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2875bf361ffdf60480ba06dfcd4e7404ecd888d2d1db48278867d245183cee6d","family_id":"qf-density-derivative-sum-coefficients","seed":202610094086,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,10,12,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 10, 12, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[11, 26, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [11, 26, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0087","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66005ebccea7aa1a713026ef191cc98843e86540bca30f1feb189ba01958a852","family_id":"qf-density-derivative-sum-coefficients","seed":202610094087,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,11,13,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 11, 13, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[12, 28, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [12, 28, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0088","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bc2695ccf766125525846b28483e53376d7fa4ddf15b6a208047e8e3fe7ab1d","family_id":"qf-density-derivative-sum-coefficients","seed":202610094088,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,12,14,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 12, 14, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[13, 30, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [13, 30, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0089","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"febb809300be8d4934ddcb0c2bcf60b407106d55c1cef9d6a49cd5e59586ff68","family_id":"qf-density-derivative-sum-coefficients","seed":202610094089,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,13,15,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 13, 15, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[14, 32, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [14, 32, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0090","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1cbc7056dfb41c619e1c170662a54a7d99cd397fe2cb66dcb02eecb3af30a8e","family_id":"qf-density-derivative-sum-coefficients","seed":202610094090,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,14,16,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 14, 16, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[15, 34, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [15, 34, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0091","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db68c5ee6af80699de104528d659e6da4465023c1ea89291890eb4218992c680","family_id":"qf-density-derivative-sum-coefficients","seed":202610094091,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,15,17,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 15, 17, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[16, 36, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [16, 36, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0092","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90efcaf1f8240f68043e743cd1b63da48aec06e8bfeba45a81460e666e87fb64","family_id":"qf-density-derivative-sum-coefficients","seed":202610094092,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,16,18,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 16, 18, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[17, 38, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [17, 38, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0093","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"550fda3426439fbef4cc4f0fdca4b265a59f51bfb12387d1ce81d804d4bac318","family_id":"qf-density-derivative-sum-coefficients","seed":202610094093,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,17,19,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 17, 19, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[18, 40, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [18, 40, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0094","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48e69d9a0258fd766b840c09334326c51b71664166e7e3b2d18f4eb35ea0d753","family_id":"qf-density-derivative-sum-coefficients","seed":202610094094,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,18,20,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 18, 20, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[19, 42, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [19, 42, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0095","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0072d06d0b111f12f8d78987897ada1dbc4ac17fbbf36b6578be76da1fceff8e","family_id":"qf-density-derivative-sum-coefficients","seed":202610094095,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,19,21,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 19, 21, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[20, 44, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [20, 44, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0096","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26e6b338f8b392e165316eebaa873112d4b990b7b82cf4d0c83c9fc8064fb185","family_id":"qf-density-derivative-sum-coefficients","seed":202610094096,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,20,22,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 20, 22, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[21, 46, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [21, 46, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0097","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9831ca33a013ac58ca8130b97640234a3d911b296d4c762c85cf40dca8b14fb","family_id":"qf-density-derivative-sum-coefficients","seed":202610094097,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,21,23,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 21, 23, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[22, 48, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [22, 48, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0098","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e47e13e71a620dcb75666ca8b5b335d26a3442f2821ed5a0cb8e1c1a1aa4e758","family_id":"qf-density-derivative-sum-coefficients","seed":202610094098,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,22,24,2]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 22, 24, 2], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[23, 50, 9]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [23, 50, 9]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0099","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"451ae300fcfac5581bb96cda662bc0c78393fc14d69aa3430e06a7c51d8ddd17","family_id":"qf-density-derivative-sum-coefficients","seed":202610094099,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,23,25,3]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 23, 25, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[24, 52, 12]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [24, 52, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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-20261009-r04-derivative-rules-0100","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"342d6bae8ac507642ddae601ffb77380602b1f9e87776a66357aa4cf8f928ff6","family_id":"qf-density-derivative-sum-coefficients","seed":202610094100,"semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,24,26,1]]"},"fields":{"title":"Complete Ascending Derivative Coefficients of a Sum","stem":"On all R, f has ascending coefficients [1, 24, 26, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.","answer":{"kind":"expression","value":"[25, 54, 6]","explanation":"Add coefficientwise, differentiate, and remove trailing zero terms to obtain [25, 54, 6]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."},"skills":["Complete Ascending Derivative Coefficients of a 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."}}