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Question 29955: Derivative Coefficients of a Rational Candidate

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0058-29955 · q-20261007-advance-20261009-r05-antiderivative-0080

Original question and givens

On all R, F has ascending rational coefficients [-8/1, 5/2, -1/1, 3/2, 4/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.

  • G001 — On all R, F has ascending rational coefficients [-8/1, 5/2, -1/1, 3/2, 4/5]. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001

S002 · implies

Before

source roles = [candidate F, [[-8, 1], [5, 2], [-1, 1], [3, 2], [4, 5]], Yes]

After

derived = [[-8/1, 5/2, -1/1, 3/2, 4/5]]

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 dependencies

Rule: Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero · Uses: G001, S001

S003 · calculation

Before

Primitive candidate ORIGINAL derivative degree0 := mul(1, 5/2)

After

Primitive candidate ORIGINAL derivative degree0 = 5/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Primitive candidate ORIGINAL derivative degree1 := mul(2, -1)

After

Primitive candidate ORIGINAL derivative degree1 = -2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Primitive candidate ORIGINAL derivative degree2 := mul(3, 3/2)

After

Primitive candidate ORIGINAL derivative degree2 = 9/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Primitive candidate ORIGINAL derivative degree3 := mul(4, 4/5)

After

Primitive candidate ORIGINAL derivative degree3 = 16/5

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [[-8/1, 5/2, -1/1, 3/2, 4/5]]

After

derived = [[5/2, -2/1, 9/2, 16/5], [5/2, -2/1, 9/2, 16/5]]

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 dependencies

Rule: 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

[5/2, -2/1, 9/2, 16/5]

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: 1a5a5a8ac9e0d91c0a496fd067b780d27e5236ab9bcf9111373c658299c0be71
  • candidate record SHA-256: 27f78cbd5dedd19c7bc107b387e5fd91a29964ca4665507fd453f0e295b457f8
  • semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df
  • question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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