# Question 3627: Eligibility of a Candidate Product-Derivative Value [Return to catalogue page 19](https://ocme.evemisslab.com/en/questions/page/19/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r04-derivative-rules-foundations/#q-20261007-advance-20261009-r04-derivative-rules-0027) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0006-03627 · q-20261007-advance-20261009-r04-derivative-rules-0027 ## Original question and givens 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? - G001 — 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? ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-derivative-rules](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) ## Step-by-step derivation S001 · impliesBefore f ascending coefficients=[5] After real polynomials are differentiable;use coefficient identities Conditions: none Coefficient k multiplies x^k;real polynomials differentiate termwise and constants have derivative0.Coefficient identities and a candidate comparison at one point are different claims. Step identifiers and dependenciesRule: polynomial_derivative_premise · Uses: G001 S002 · impliesBefore f(x) ascending coefficients=[5] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S001 S003 · impliesBefore g(x) ascending coefficients=[-2] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S002 S004 · impliesBefore f′(x) ascending coefficients=[0] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S003 S005 · impliesBefore g′(x) ascending coefficients=[0] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S004 S006 · calculationBefore f′(x)g(x) := mul(0, -2) After f′(x)g(x) = 0 Conditions: none First product-rule term:differentiate f and retain g. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore f(x)g′(x) := mul(5, 0) After f(x)g′(x) = 0 Conditions: none The second term differentiates only g;do not use f′g′. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore product derivative := sum(0, 0) After product derivative = 0 Conditions: none Add both terms to obtain(fg)′(x). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore candidate=1;actual product derivative=0 After No Conditions: none Compare the candidate at this specified x;the differentiation rule rests on polynomial differentiability,not a one-point coincidence. Step identifiers and dependenciesRule: product_rule_candidate_comparison · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute product derivative with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute f(x)g′(x) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute f′(x)g(x) with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result No The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2894081a942b26e3277c08295c5ba85e87101a248db5981026b51377b360baa4 - candidate record SHA-256: 8e741e2fc7355193cb5a5b64bf2ea74d4d0ac61525d7f8546a696a12b36b0a7b - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r04-derivative-rules-0027.json)