# Question 3601: 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-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. R0006-03601 · q-20261007-advance-20261009-r04-derivative-rules-0051 ## Original question and givens On all R, f has ascending coefficients [3], constant term first and g has ascending coefficients [1, 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 [3], constant term first and g has ascending coefficients [1, 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=[3] 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 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, S001 S003 · impliesBefore f(x) ascending coefficients=[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, S002 S004 · impliesBefore g(x) ascending coefficients=[1, 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, S003 S005 · calculationBefore g(x) Horner multiply before degree0 := mul(1, 2) After g(x) Horner multiply before degree0 = 2 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore g(x) Horner add degree0 := sum(2, 1) After g(x) Horner add degree0 = 3 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore f′(x) ascending coefficients=[0] 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, S006 S008 · impliesBefore g′(x) ascending coefficients=[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, S007 S009 · impliesBefore f′(g(x)) ascending coefficients=[0] After evaluate at x=3 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, S008 S010 · impliesBefore g′(f(x)) ascending coefficients=[1] After evaluate at x=3 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, S009 S011 · calculationBefore (f∘g)′(x) := mul(0, 1) After (f∘g)′(x) = 0 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, S010 S012 · calculationBefore (g∘f)′(x) := mul(1, 0) After (g∘f)′(x) = 0 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, S011 S013 · impliesBefore inner values=(3,3);inner derivatives=(1,0) After ordered derivative values=(0/1, 0/1) 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, S012 ## Reverse checks - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute (g∘f)′(x) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute (f∘g)′(x) with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - 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 — Recompute g′ 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 (0/1, 0/1) 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: 33a73407c3cd04826d750b831dcdf74a15fe0e4b8b1358fee268db4575aa7f09 - candidate record SHA-256: 697519cad34f66a1dda48c813fba9a6563920941f49d3d295380fcad4de2720f - 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-0051.json)