# Question 31132: Chain-Rule Derivative of a Polynomial Composition

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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.

R0061-31132 · q-20261007-advance-20261008-r05-derivative-rules-0082



## Original question and givens

On all R, f has ascending coefficients [-6, 3, 0, 1], starting with its constant term; g has ascending coefficients [1, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.


- G001 — On all R, f has ascending coefficients [-6, 3, 0, 1], starting with its constant term; g has ascending coefficients [1, 1, 1, 1], starting with its constant term. Find the derivative of f∘g at x=1. Input degrees are at most three; retain all higher terms produced by the operation.





## 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

source roles = [qf-derivative-rule-chain, derivative_chain, domain, reals]

After

derived = [[integer_derivative_value]]

Conditions: none

Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.

Step identifiers and dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001





S002 · impliesBefore

source roles = [f OUTER or first, [-6, 3, 0, 1]]

After

derived = [[-6, 3, 0, 1]]

Conditions: none

Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability.

Step identifiers and dependenciesRule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S001





S003 · impliesBefore

source roles = [g INNER or second, [1, 1, 1, 1]]

After

derived = [[1, 1, 1, 1]]

Conditions: none

Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability.

Step identifiers and dependenciesRule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S002





S004 · calculationBefore

f derivative coefficient power1 := mul(1, 3)

After

f derivative coefficient power1 = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

f derivative coefficient power2 := mul(2, 0)

After

f derivative coefficient power2 = 0

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

f derivative coefficient power3 := mul(3, 1)

After

f derivative coefficient power3 = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

source roles = [f, [-6, 3, 0, 1]]

After

derived = [[3, 0, 3]]

Conditions: none

For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds.

Step identifiers and dependenciesRule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S006





S008 · calculationBefore

g derivative coefficient power1 := mul(1, 1)

After

g derivative coefficient power1 = 1

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

g derivative coefficient power2 := mul(2, 1)

After

g derivative coefficient power2 = 2

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

g derivative coefficient power3 := mul(3, 1)

After

g derivative coefficient power3 = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

source roles = [g, [1, 1, 1, 1]]

After

derived = [[1, 2, 3]]

Conditions: none

For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds.

Step identifiers and dependenciesRule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S010





S012 · calculationBefore

g derivative at ORIGINAL x Horner power2 times argument := mul(0, 1)

After

g derivative at ORIGINAL x Horner power2 times argument = 0

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

g derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 3)

After

g derivative at ORIGINAL x Horner power2 plus coefficient = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

g derivative at ORIGINAL x Horner power1 times argument := mul(3, 1)

After

g derivative at ORIGINAL x Horner power1 times argument = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

g derivative at ORIGINAL x Horner power1 plus coefficient := sum(3, 2)

After

g derivative at ORIGINAL x Horner power1 plus coefficient = 5

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

g derivative at ORIGINAL x Horner power0 times argument := mul(5, 1)

After

g derivative at ORIGINAL x Horner power0 times argument = 5

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

g derivative at ORIGINAL x Horner power0 plus coefficient := sum(5, 1)

After

g derivative at ORIGINAL x Horner power0 plus coefficient = 6

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

source roles = [g derivative at ORIGINAL x, [1, 2, 3], 1]

After

derived = [6]

Conditions: none

Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x.

Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S017





S019 · calculationBefore

g INNER at ORIGINAL x Horner power3 times argument := mul(0, 1)

After

g INNER at ORIGINAL x Horner power3 times argument = 0

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

g INNER at ORIGINAL x Horner power3 plus coefficient := sum(0, 1)

After

g INNER at ORIGINAL x Horner power3 plus coefficient = 1

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

g INNER at ORIGINAL x Horner power2 times argument := mul(1, 1)

After

g INNER at ORIGINAL x Horner power2 times argument = 1

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

g INNER at ORIGINAL x Horner power2 plus coefficient := sum(1, 1)

After

g INNER at ORIGINAL x Horner power2 plus coefficient = 2

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

g INNER at ORIGINAL x Horner power1 times argument := mul(2, 1)

After

g INNER at ORIGINAL x Horner power1 times argument = 2

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

g INNER at ORIGINAL x Horner power1 plus coefficient := sum(2, 1)

After

g INNER at ORIGINAL x Horner power1 plus coefficient = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

g INNER at ORIGINAL x Horner power0 times argument := mul(3, 1)

After

g INNER at ORIGINAL x Horner power0 times argument = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

g INNER at ORIGINAL x Horner power0 plus coefficient := sum(3, 1)

After

g INNER at ORIGINAL x Horner power0 plus coefficient = 4

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · impliesBefore

source roles = [g INNER at ORIGINAL x, [1, 1, 1, 1], 1]

After

derived = [4]

Conditions: none

Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x.

Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S026





S028 · calculationBefore

f derivative at INNER g(x) Horner power2 times argument := mul(0, 4)

After

f derivative at INNER g(x) Horner power2 times argument = 0

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

f derivative at INNER g(x) Horner power2 plus coefficient := sum(0, 3)

After

f derivative at INNER g(x) Horner power2 plus coefficient = 3

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

f derivative at INNER g(x) Horner power1 times argument := mul(3, 4)

After

f derivative at INNER g(x) Horner power1 times argument = 12

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

f derivative at INNER g(x) Horner power1 plus coefficient := sum(12, 0)

After

f derivative at INNER g(x) Horner power1 plus coefficient = 12

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

f derivative at INNER g(x) Horner power0 times argument := mul(12, 4)

After

f derivative at INNER g(x) Horner power0 times argument = 48

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

f derivative at INNER g(x) Horner power0 plus coefficient := sum(48, 3)

After

f derivative at INNER g(x) Horner power0 plus coefficient = 51

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

source roles = [f derivative at INNER g(x), [3, 0, 3], 4]

After

derived = [51]

Conditions: none

Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x.

Step identifiers and dependenciesRule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S033





S035 · calculationBefore

Derivative chain ORIGINAL outer derivative at gx times inner derivative := mul(51, 6)

After

Derivative chain ORIGINAL outer derivative at gx times inner derivative = 306

Conditions: none

Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

source roles = [[-6, 3, 0, 1], [1, 1, 1, 1], 1, [3, 0, 3], [1, 2, 3]]

After

derived = [4, 51, 6, 306]

Conditions: none

For h=f∘g,FIRST evaluate g(x),THEN f′(g(x)),FINALLY multiply g′(x).This is neither f′(x)g′(x)nor reversed g∘f;whole polynomials and constants are allowed.

Step identifiers and dependenciesRule: Derivative_CHAIN_outer_f_inner_g_argument_AND_inner_factor · Uses: G001, S035









## Reverse checks


- S036 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S035 · premise_and_result_audit — Recompute Derivative chain ORIGINAL outer derivative at gx times inner derivative with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S033 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power0 times argument with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power1 times argument with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute f derivative at INNER g(x) Horner power2 times argument with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S026 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power3 plus coefficient with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute g INNER at ORIGINAL x Horner power3 times argument with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S017 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 plus coefficient with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power0 times argument with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power1 plus coefficient with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power1 times argument with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power2 plus coefficient with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute g derivative at ORIGINAL x Horner power2 times argument with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S010 · premise_and_result_audit — Recompute g derivative coefficient power3 with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute g derivative coefficient power2 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute g derivative coefficient power1 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S006 · premise_and_result_audit — Recompute f derivative coefficient power3 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute f derivative coefficient power2 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute f derivative coefficient power1 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
- S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.





## Result

306

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: e2d6deff71578efa18dba2095921fe79cceec91b5ece2458699b4b772109585d
- candidate record SHA-256: 2c5c4b6f64160e23a8bff46783101f882f32b8a2dd44676f524ca7a4494f7599
- semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69
- question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495
- Lean: NOT_RUN · human review: NOT_PERFORMED

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