# Question 3624: Derivatives in Both Composition Orders [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-0074) 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. R0006-03624 · q-20261007-advance-20261009-r04-derivative-rules-0074 ## Original question and givens 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. - G001 — 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. ## 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=[11, 24, 1, 4] 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, 24) After f′ coefficient at degree0 = 24 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, 4) After f′ coefficient at degree2 = 12 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, 3) After g′ coefficient at degree1 = 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, S005 S007 · calculationBefore g′ coefficient at degree2 := mul(3, 1) After g′ coefficient at degree2 = 3 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=[11, 24, 1, 4] After evaluate at x=2 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(4, 2) After f(x) Horner multiply before degree2 = 8 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(8, 1) After f(x) Horner add degree2 = 9 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(9, 2) After f(x) Horner multiply before degree1 = 18 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(18, 24) After f(x) Horner add degree1 = 42 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(42, 2) After f(x) Horner multiply before degree0 = 84 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(84, 11) After f(x) Horner add degree0 = 95 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=[18, 1, 3, 1] After evaluate at x=2 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(1, 2) After g(x) Horner multiply before degree2 = 2 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(2, 3) After g(x) Horner add degree2 = 5 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(5, 2) After g(x) Horner multiply before degree1 = 10 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(10, 1) After g(x) Horner add degree1 = 11 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(11, 2) After g(x) Horner multiply before degree0 = 22 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(22, 18) After g(x) Horner add degree0 = 40 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=[24, 2, 12] After evaluate at x=2 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(12, 2) After f′(x) Horner multiply before degree1 = 24 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(24, 2) After f′(x) Horner add degree1 = 26 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(26, 2) After f′(x) Horner multiply before degree0 = 52 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(52, 24) After f′(x) Horner add degree0 = 76 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, 6, 3] After evaluate at x=2 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(3, 2) After g′(x) Horner multiply before degree1 = 6 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(6, 6) After g′(x) Horner add degree1 = 12 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(12, 2) After g′(x) Horner multiply before degree0 = 24 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(24, 1) After g′(x) Horner add degree0 = 25 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore f′(g(x)) ascending coefficients=[24, 2, 12] After evaluate at x=40 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, S031 S033 · calculationBefore f′(g(x)) Horner multiply before degree1 := mul(12, 40) After f′(g(x)) Horner multiply before degree1 = 480 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore f′(g(x)) Horner add degree1 := sum(480, 2) After f′(g(x)) Horner add degree1 = 482 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore f′(g(x)) Horner multiply before degree0 := mul(482, 40) After f′(g(x)) Horner multiply before degree0 = 19280 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore f′(g(x)) Horner add degree0 := sum(19280, 24) After f′(g(x)) Horner add degree0 = 19304 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore g′(f(x)) ascending coefficients=[1, 6, 3] After evaluate at x=95 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, S036 S038 · calculationBefore g′(f(x)) Horner multiply before degree1 := mul(3, 95) After g′(f(x)) Horner multiply before degree1 = 285 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore g′(f(x)) Horner add degree1 := sum(285, 6) After g′(f(x)) Horner add degree1 = 291 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore g′(f(x)) Horner multiply before degree0 := mul(291, 95) After g′(f(x)) Horner multiply before degree0 = 27645 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore g′(f(x)) Horner add degree0 := sum(27645, 1) After g′(f(x)) Horner add degree0 = 27646 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore (f∘g)′(x) := mul(19304, 25) After (f∘g)′(x) = 482600 Conditions: none Evaluate the outer derivative f′ at g(x),then multiply by the inner derivative g′(x). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore (g∘f)′(x) := mul(27646, 76) After (g∘f)′(x) = 2101096 Conditions: none The reverse order changes the outer function,the evaluation point and the inner derivative. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore inner values=(40,95);inner derivatives=(25,76) After ordered derivative values=(482600/1, 2101096/1) Conditions: none The entries are f after g and g after f in that order;do not swap them.A constant outer function naturally makes its corresponding composite derivative0. Step identifiers and dependenciesRule: chain_rule_order_guard · Uses: G001, S043 ## Reverse checks - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Recompute (g∘f)′(x) with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute (f∘g)′(x) with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (482600/1, 2101096/1) 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: 8bed912e9b57e100480c4ea6bf66f53313e81b9e4f5f2d8648b386cf7c8dad41 - candidate record SHA-256: b27262df5b9ff9624952e20e7b1b93038f6350d440b8c8474785804d6fae1f69 - 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-0074.json)