# Question 31164: 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-0064) 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. R0061-31164 · q-20261007-advance-20261008-r05-derivative-rules-0064 ## Original question and givens On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation. - G001 — On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term; g has ascending coefficients [2, 0, 1, 3], starting with its constant term. Find the derivative of fg at x=1. Input degrees are at most three; retain all higher terms produced by the operation. ## 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, [1, 14, 0, 2]] After derived = [[1, 14, 0, 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, S001 S003 · impliesBefore source roles = [g INNER or second, [2, 0, 1, 3]] After derived = [[2, 0, 1, 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, S002 S004 · calculationBefore f derivative coefficient power1 := mul(1, 14) After f derivative coefficient power1 = 14 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, S003 S005 · calculationBefore f derivative coefficient power2 := mul(2, 0) After f derivative coefficient power2 = 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, S004 S006 · calculationBefore f derivative coefficient power3 := mul(3, 2) After f derivative coefficient power3 = 6 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 · impliesBefore source roles = [f, [1, 14, 0, 2]] After derived = [[14, 0, 6]] 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, S006 S008 · calculationBefore g derivative coefficient power1 := mul(1, 0) After g derivative coefficient power1 = 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, S007 S009 · calculationBefore g derivative coefficient power2 := mul(2, 1) After g derivative coefficient power2 = 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, S008 S010 · calculationBefore g derivative coefficient power3 := mul(3, 3) After g derivative coefficient power3 = 9 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 = [g, [2, 0, 1, 3]] After derived = [[0, 2, 9]] 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, S010 S012 · calculationBefore g derivative at ORIGINAL x Horner power2 times argument := mul(0, 1) After g derivative at ORIGINAL x Horner power2 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 g derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 9) After g derivative at ORIGINAL x Horner power2 plus coefficient = 9 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 · calculationBefore g derivative at ORIGINAL x Horner power1 times argument := mul(9, 1) After g derivative at ORIGINAL x Horner power1 times argument = 9 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, S013 S015 · calculationBefore g derivative at ORIGINAL x Horner power1 plus coefficient := sum(9, 2) After g derivative at ORIGINAL x Horner power1 plus coefficient = 11 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 derivative at ORIGINAL x Horner power0 times argument := mul(11, 1) After g derivative at ORIGINAL x Horner power0 times argument = 11 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 · calculationBefore g derivative at ORIGINAL x Horner power0 plus coefficient := sum(11, 0) After g derivative at ORIGINAL x Horner power0 plus coefficient = 11 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, S016 S018 · impliesBefore source roles = [g derivative at ORIGINAL x, [0, 2, 9], 1] After derived = [11] 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, S017 S019 · calculationBefore f derivative at ORIGINAL x Horner power2 times argument := mul(0, 1) After f derivative at ORIGINAL x Horner power2 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, S018 S020 · calculationBefore f derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 6) After f derivative at ORIGINAL x Horner power2 plus coefficient = 6 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 · calculationBefore f derivative at ORIGINAL x Horner power1 times argument := mul(6, 1) After f derivative at ORIGINAL x Horner power1 times argument = 6 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, S020 S022 · calculationBefore f derivative at ORIGINAL x Horner power1 plus coefficient := sum(6, 0) After f derivative at ORIGINAL x Horner power1 plus coefficient = 6 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, S021 S023 · calculationBefore f derivative at ORIGINAL x Horner power0 times argument := mul(6, 1) After f derivative at ORIGINAL x Horner power0 times argument = 6 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, S022 S024 · calculationBefore f derivative at ORIGINAL x Horner power0 plus coefficient := sum(6, 14) After f derivative at ORIGINAL x Horner power0 plus coefficient = 20 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, S023 S025 · impliesBefore source roles = [f derivative at ORIGINAL x, [14, 0, 6], 1] After derived = [20] 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, S024 S026 · calculationBefore f value at ORIGINAL x Horner power3 times argument := mul(0, 1) After f value at ORIGINAL x Horner power3 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, S025 S027 · calculationBefore f value at ORIGINAL x Horner power3 plus coefficient := sum(0, 2) After f value at ORIGINAL x Horner power3 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, S026 S028 · calculationBefore f value at ORIGINAL x Horner power2 times argument := mul(2, 1) After f value at ORIGINAL x Horner power2 times argument = 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, S027 S029 · calculationBefore f value at ORIGINAL x Horner power2 plus coefficient := sum(2, 0) After f value at ORIGINAL x Horner power2 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, S028 S030 · calculationBefore f value at ORIGINAL x Horner power1 times argument := mul(2, 1) After f value at ORIGINAL x Horner power1 times argument = 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, S029 S031 · calculationBefore f value at ORIGINAL x Horner power1 plus coefficient := sum(2, 14) After f value at ORIGINAL x Horner power1 plus coefficient = 16 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, S030 S032 · calculationBefore f value at ORIGINAL x Horner power0 times argument := mul(16, 1) After f value at ORIGINAL x Horner power0 times argument = 16 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, S031 S033 · calculationBefore f value at ORIGINAL x Horner power0 plus coefficient := sum(16, 1) After f value at ORIGINAL x Horner power0 plus coefficient = 17 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, S032 S034 · impliesBefore source roles = [f value at ORIGINAL x, [1, 14, 0, 2], 1] After derived = [17] 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, S033 S035 · calculationBefore g value at ORIGINAL x Horner power3 times argument := mul(0, 1) After g value at ORIGINAL x Horner power3 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, S034 S036 · calculationBefore g value at ORIGINAL x Horner power3 plus coefficient := sum(0, 3) After g value at ORIGINAL x Horner power3 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, S035 S037 · calculationBefore g value at ORIGINAL x Horner power2 times argument := mul(3, 1) After g value at ORIGINAL x Horner power2 times argument = 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, S036 S038 · calculationBefore g value at ORIGINAL x Horner power2 plus coefficient := sum(3, 1) After g value at ORIGINAL x Horner power2 plus coefficient = 4 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, S037 S039 · calculationBefore g value at ORIGINAL x Horner power1 times argument := mul(4, 1) After g value at ORIGINAL x Horner power1 times argument = 4 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, S038 S040 · calculationBefore g value at ORIGINAL x Horner power1 plus coefficient := sum(4, 0) After g value at ORIGINAL x Horner power1 plus coefficient = 4 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, S039 S041 · calculationBefore g value at ORIGINAL x Horner power0 times argument := mul(4, 1) After g value at ORIGINAL x Horner power0 times argument = 4 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, S040 S042 · calculationBefore g value at ORIGINAL x Horner power0 plus coefficient := sum(4, 2) After g value at ORIGINAL x Horner power0 plus coefficient = 6 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, S041 S043 · impliesBefore source roles = [g value at ORIGINAL x, [2, 0, 1, 3], 1] After derived = [6] 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, S042 S044 · calculationBefore Derivative product ORIGINAL fprime times g := mul(20, 6) After Derivative product ORIGINAL fprime times g = 120 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, S043 S045 · calculationBefore Derivative product ORIGINAL f times gprime := mul(17, 11) After Derivative product ORIGINAL f times gprime = 187 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, S044 S046 · calculationBefore Derivative product ORIGINAL two terms sum := sum(120, 187) After Derivative product ORIGINAL two terms sum = 307 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, S045 S047 · impliesBefore source roles = [[1, 14, 0, 2], [2, 0, 1, 3], 1, [14, 0, 6], [0, 2, 9]] After derived = [17, 6, 120, 187, 307] 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, S046 ## Reverse checks - S047 · 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. - S046 · premise_and_result_audit — Recompute Derivative product ORIGINAL two terms sum with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Derivative product ORIGINAL f times gprime with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Derivative product ORIGINAL fprime times g with exact rational arithmetic, not rounded decimals. - S043 · 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. - S042 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power3 plus coefficient with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute g value at ORIGINAL x Horner power3 times argument with exact rational arithmetic, not rounded decimals. - S034 · 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. - S033 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power3 plus coefficient with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute f value at ORIGINAL x Horner power3 times argument with exact rational arithmetic, not rounded decimals. - S025 · 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. - S024 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute f derivative at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals. - S018 · 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. - S017 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power2 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 g derivative coefficient power3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute g derivative coefficient power2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute g derivative coefficient power1 with exact rational arithmetic, not rounded decimals. - S007 · 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. - S006 · premise_and_result_audit — Recompute f derivative coefficient power3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute f derivative coefficient power2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute f derivative coefficient power1 with exact rational arithmetic, not rounded decimals. - 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 307 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: 42fe75d133bd107c8e8e21b1691cba6fe1b1b35127637feff00ab2cf4400eb39 - candidate record SHA-256: e8743723ee29e348463ba752aa52b82670dbdc4206fd922d50b51d705bccc664 - 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-0064.json)