# Question 3685: Complete Ascending Derivative Coefficients of a Sum [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-0085) 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. R0006-03685 · q-20261007-advance-20261009-r04-derivative-rules-0085 ## Original question and givens On all R, f has ascending coefficients [1, 9, 11, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′. - G001 — On all R, f has ascending coefficients [1, 9, 11, 1], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′. ## 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=[1, 9, 11, 1] 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, 9) After f′ coefficient at degree0 = 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, S001 S003 · calculationBefore f′ coefficient at degree1 := mul(2, 11) After f′ coefficient at degree1 = 22 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, 1) After f′ 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, 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, 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 · calculationBefore sum derivative coefficient0 := sum(9, 1) After sum derivative coefficient0 = 10 Conditions: none Add derivative coefficients of the same degree,using0 for a missing high-degree term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore sum derivative coefficient1 := sum(22, 2) After sum derivative coefficient1 = 24 Conditions: none Add derivative coefficients of the same degree,using0 for a missing high-degree term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore sum derivative coefficient2 := sum(3, 3) After sum derivative coefficient2 = 6 Conditions: none Add derivative coefficients of the same degree,using0 for a missing high-degree term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore [10, 24, 6] After complete derivative coefficient identity;zero polynomial=[0] Conditions: none The sum rule gives(f+g)′=f′+g′.Removing trailing zeros preserves the polynomial;retain[0] for zero.This is not coefficient inference from a few sample points. Step identifiers and dependenciesRule: sum_rule_coefficient_identity · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute sum derivative coefficient2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute sum derivative coefficient1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute sum derivative coefficient0 with exact rational arithmetic, not rounded decimals. - 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 [10, 24, 6] 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: ffe23302f491eb2209be5840634495bbbdcc9f44c923ee9d2b9f57b6ddafe4b3 - candidate record SHA-256: 00acfe05d0c9867ab5017362467a44e34904970c396687c91d47d1371b7003ba - 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-0085.json)