# Question 31151: Derivative of a Polynomial Product [Return to catalogue page 156](https://ocme.evemisslab.com/en/questions/page/156/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r05-derivative-rules-foundations/#q-20261007-advance-20261008-r05-derivative-rules-0051) 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. R0061-31151 · q-20261007-advance-20261008-r05-derivative-rules-0051 ## Original question and givens On all R, f has ascending coefficients [3], starting with its constant term; g has ascending coefficients [-2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation. - G001 — On all R, f has ascending coefficients [3], starting with its constant term; g has ascending coefficients [-2], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation. ## 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 source roles = [qf-derivative-rule-product, derivative_product, domain, reals] After derived = [[integer_derivative_value]] Conditions: none Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status. Step identifiers and dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001 S002 · impliesBefore source roles = [f OUTER or first, [3]] After derived = [[3]] Conditions: none Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability. Step identifiers and dependenciesRule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S001 S003 · impliesBefore source roles = [g INNER or second, [-2]] After derived = [[-2]] Conditions: none Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability. Step identifiers and dependenciesRule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S002 S004 · impliesBefore source roles = [f, [3]] After derived = [[0]] Conditions: none For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds. Step identifiers and dependenciesRule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S003 S005 · impliesBefore source roles = [g, [-2]] After derived = [[0]] Conditions: none For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds. Step identifiers and dependenciesRule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S004 S006 · calculationBefore g derivative at ORIGINAL x Horner power0 times argument := mul(0, 1) After g derivative at ORIGINAL x Horner power0 times argument = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore g derivative at ORIGINAL x Horner power0 plus coefficient := sum(0, 0) After g derivative at ORIGINAL x Horner power0 plus coefficient = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [g derivative at ORIGINAL x, [0], 1] After derived = [0] Conditions: none Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x. Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S007 S009 · calculationBefore f derivative at ORIGINAL x Horner power0 times argument := mul(0, 1) After f derivative at ORIGINAL x Horner power0 times argument = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore f derivative at ORIGINAL x Horner power0 plus coefficient := sum(0, 0) After f derivative at ORIGINAL x Horner power0 plus coefficient = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [f derivative at ORIGINAL x, [0], 1] After derived = [0] Conditions: none Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x. Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S010 S012 · calculationBefore f value at ORIGINAL x Horner power0 times argument := mul(0, 1) After f value at ORIGINAL x Horner power0 times argument = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore f value at ORIGINAL x Horner power0 plus coefficient := sum(0, 3) After f value at ORIGINAL x Horner power0 plus coefficient = 3 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [f value at ORIGINAL x, [3], 1] After derived = [3] Conditions: none Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x. Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S013 S015 · calculationBefore g value at ORIGINAL x Horner power0 times argument := mul(0, 1) After g value at ORIGINAL x Horner power0 times argument = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore g value at ORIGINAL x Horner power0 plus coefficient := sum(0, -2) After g value at ORIGINAL x Horner power0 plus coefficient = -2 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [g value at ORIGINAL x, [-2], 1] After derived = [-2] Conditions: none Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x. Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S016 S018 · calculationBefore Derivative product ORIGINAL fprime times g := mul(0, -2) After Derivative product ORIGINAL fprime times g = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Derivative product ORIGINAL f times gprime := mul(3, 0) After Derivative product ORIGINAL f times gprime = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Derivative product ORIGINAL two terms sum := sum(0, 0) After Derivative product ORIGINAL two terms sum = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [[3], [-2], 1, [0], [0]] After derived = [3, -2, 0, 0, 0] Conditions: none The product rule is(fg)′=f′g+fg′;retain BOTH terms,not f′g′.Evaluate at the SAME x without raw degree/value caps on derived results. Step identifiers and dependenciesRule: Derivative_PRODUCT_rule_BOTH_terms_NOT_product_of_derivatives · Uses: G001, S020 ## Reverse checks - S021 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S020 · premise_and_result_audit — Recompute Derivative product ORIGINAL two terms sum with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Derivative product ORIGINAL f times gprime with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Derivative product ORIGINAL fprime times g with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S016 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S013 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S010 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S007 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. ## Result 0 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: 7ad009e6e992f40037a0c110955d3e5c1bfd4c4aabea116269a71470335174a2 - candidate record SHA-256: 25cf8bbe6939014a1684acdaa782ec4c015f2b62fce226afe7dee647faf00ac4 - semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69 - question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r05-derivative-rules-0051.json)