# Question 29972: Derivative Coefficients of a Rational Candidate [Return to catalogue page 150](https://ocme.evemisslab.com/en/questions/page/150/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r05-antiderivative-foundations/#q-20261007-advance-20261009-r05-antiderivative-0097) 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. R0058-29972 · q-20261007-advance-20261009-r05-antiderivative-0097 ## Original question and givens On all R, F has ascending rational coefficients [9/1, 11/1, -20/3, 23/4, -21/5]. Give its complete normalized ascending derivative coefficients as reduced fractions. - G001 — On all R, F has ascending rational coefficients [9/1, 11/1, -20/3, 23/4, -21/5]. Give its complete normalized ascending derivative coefficients as reduced fractions. ## 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-antiderivative](https://ocme.evemisslab.com/en/mko/mko-antiderivative/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density-primitive-candidate-derivative, density_primitive_candidate_derivative] After derived = [[ALL_normalized_ascending_derivative_coefficients_string_retaining1]] Conditions: none Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention. Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001 S002 · impliesBefore source roles = [candidate F, [[9, 1], [11, 1], [-20, 3], [23, 4], [-21, 5]], Yes] After derived = [[9/1, 11/1, -20/3, 23/4, -21/5]] Conditions: none Raw integer f has length1..4 and rational candidate F length1..5.Every raw number is SAFE;rational denominators are STRICTLY positive,not repaired from negatives.Trim only trailing zeros,keeping a singleton zero polynomial.Derived algebra has no1e6 or raw-degree cap. Step identifiers and dependenciesRule: Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero · Uses: G001, S001 S003 · calculationBefore Primitive candidate ORIGINAL derivative degree0 := mul(1, 11) After Primitive candidate ORIGINAL derivative degree0 = 11 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Primitive candidate ORIGINAL derivative degree1 := mul(2, -20/3) After Primitive candidate ORIGINAL derivative degree1 = -40/3 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Primitive candidate ORIGINAL derivative degree2 := mul(3, 23/4) After Primitive candidate ORIGINAL derivative degree2 = 69/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Primitive candidate ORIGINAL derivative degree3 := mul(4, -21/5) After Primitive candidate ORIGINAL derivative degree3 = -84/5 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [[9/1, 11/1, -20/3, 23/4, -21/5]] After derived = [[11/1, -40/3, 69/4, -84/5], [11/1, -40/3, 69/4, -84/5]] Conditions: none Differentiate EVERY nonconstant term into i*c_i*x^(i-1),list the COMPLETE ascending coefficients,and trim trailing zeros.Constants differentiate to0,so a constant candidate returns[0/1],not[].No input f was provided;do not invent a Yes/No eligibility test F_prime=f. Step identifiers and dependenciesRule: Primitive_candidate_COMPLETE_derivative_coefficients_not_eligibility · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S006 · premise_and_result_audit — Recompute Primitive candidate ORIGINAL derivative degree3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Primitive candidate ORIGINAL derivative degree2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Primitive candidate ORIGINAL derivative degree1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Primitive candidate ORIGINAL derivative degree0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. ## Result [11/1, -40/3, 69/4, -84/5] 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: 7515dfbd11eff67fd03e4fd4c5ffd53761d14b173e3e14a9c250491740ecfd6f - candidate record SHA-256: c3fe31ad3a52f6e7f52fdaad29a41c67576d9001acae11f0358de77c47f0e5c9 - semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df - question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r05-antiderivative-0097.json)