# Question 3631: 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-0031) 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. R0006-03631 · q-20261007-advance-20261009-r04-derivative-rules-0031 ## Original question and givens 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? - G001 — 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? ## 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=[-7, 6, 1, 2] 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 · calculationBefore f′ coefficient at degree0 := mul(1, 6) After f′ coefficient at degree0 = 6 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore f′ coefficient at degree1 := mul(2, 1) After f′ coefficient at degree1 = 2 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore f′ coefficient at degree2 := mul(3, 2) After f′ coefficient at degree2 = 6 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore g′ coefficient at degree0 := mul(1, -1) After g′ coefficient at degree0 = -1 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore g′ coefficient at degree1 := mul(2, 1) After g′ coefficient at degree1 = 2 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore g′ coefficient at degree2 := mul(3, 3) After g′ coefficient at degree2 = 9 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore f(x) ascending coefficients=[-7, 6, 1, 2] After evaluate at x=0 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, S007 S009 · calculationBefore f(x) Horner multiply before degree2 := mul(2, 0) After f(x) Horner multiply before degree2 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore f(x) Horner add degree2 := sum(0, 1) After f(x) Horner add degree2 = 1 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore f(x) Horner multiply before degree1 := mul(1, 0) After f(x) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore f(x) Horner add degree1 := sum(0, 6) After f(x) Horner add degree1 = 6 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore f(x) Horner multiply before degree0 := mul(6, 0) After f(x) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore f(x) Horner add degree0 := sum(0, -7) After f(x) Horner add degree0 = -7 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore g(x) ascending coefficients=[2, -1, 1, 3] After evaluate at x=0 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, S014 S016 · calculationBefore g(x) Horner multiply before degree2 := mul(3, 0) After g(x) Horner multiply before degree2 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore g(x) Horner add degree2 := sum(0, 1) After g(x) Horner add degree2 = 1 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore g(x) Horner multiply before degree1 := mul(1, 0) After g(x) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore g(x) Horner add degree1 := sum(0, -1) After g(x) Horner add degree1 = -1 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore g(x) Horner multiply before degree0 := mul(-1, 0) After g(x) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore g(x) Horner add degree0 := sum(0, 2) After g(x) Horner add degree0 = 2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore f′(x) ascending coefficients=[6, 2, 6] After evaluate at x=0 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, S021 S023 · calculationBefore f′(x) Horner multiply before degree1 := mul(6, 0) After f′(x) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore f′(x) Horner add degree1 := sum(0, 2) After f′(x) Horner add degree1 = 2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore f′(x) Horner multiply before degree0 := mul(2, 0) After f′(x) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore f′(x) Horner add degree0 := sum(0, 6) After f′(x) Horner add degree0 = 6 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore g′(x) ascending coefficients=[-1, 2, 9] After evaluate at x=0 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, S026 S028 · calculationBefore g′(x) Horner multiply before degree1 := mul(9, 0) After g′(x) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore g′(x) Horner add degree1 := sum(0, 2) After g′(x) Horner add degree1 = 2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore g′(x) Horner multiply before degree0 := mul(2, 0) After g′(x) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore g′(x) Horner add degree0 := sum(0, -1) After g′(x) Horner add degree0 = -1 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore f′(x)g(x) := mul(6, 2) After f′(x)g(x) = 12 Conditions: none First product-rule term:differentiate f and retain g. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore f(x)g′(x) := mul(-7, -1) After f(x)g′(x) = 7 Conditions: none The second term differentiates only g;do not use f′g′. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore product derivative := sum(12, 7) After product derivative = 19 Conditions: none Add both terms to obtain(fg)′(x). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore candidate=19;actual product derivative=19 After Yes 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, S034 ## Reverse checks - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Recompute product derivative with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute f(x)g′(x) with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute f′(x)g(x) with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute g′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute g′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute f′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute f′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute g(x) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute g(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute g(x) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute g(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute f(x) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute f(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute f(x) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute f(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute f(x) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute f(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute g′ coefficient at degree2 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute g′ coefficient at degree1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute g′ coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute f′ coefficient at degree2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute f′ coefficient at degree1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute f′ coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result Yes 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: ce14b0edadff561bef2440f37fdc9a8fb33a541a9edbeef6273e1b624498c7d4 - candidate record SHA-256: 6cafaee82c586d9c8e6974973048bc736f9b11063977365077e53da0a49d8e6d - 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-0031.json)