Question 31134: Chain-Rule Derivative of a Polynomial Composition

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0061-31134 · q-20261007-advance-20261008-r05-derivative-rules-0084

Original question and givens

On all R, f has ascending coefficients [-4, 1, 2, 1], starting with its constant term; g has ascending coefficients [3, 1, 3, 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 [-4, 1, 2, 1], starting with its constant term; g has ascending coefficients [3, 1, 3, 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001

S002 · implies

Before

source roles = [f OUTER or first, [-4, 1, 2, 1]]

After

derived = [[-4, 1, 2, 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 dependencies

Rule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S001

S003 · implies

Before

source roles = [g INNER or second, [3, 1, 3, 1]]

After

derived = [[3, 1, 3, 1]]

Conditions: none

Step identifiers and dependencies

Rule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S002

S004 · calculation

Before

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

After

f 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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

After

f derivative coefficient power2 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [f, [-4, 1, 2, 1]]

After

derived = [[1, 4, 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 dependencies

Rule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

g derivative coefficient power2 := mul(2, 3)

After

g derivative coefficient power2 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [g, [3, 1, 3, 1]]

After

derived = [[1, 6, 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 dependencies

Rule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

g derivative at ORIGINAL x Horner power1 plus coefficient := sum(-3, 6)

After

g derivative 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

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

After

g derivative 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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

After

g derivative at ORIGINAL x Horner power0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [g derivative at ORIGINAL x, [1, 6, 3], -1]

After

derived = [-2]

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 dependencies

Rule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

g INNER at ORIGINAL x Horner power2 plus coefficient := sum(-1, 3)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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

After

g INNER at ORIGINAL x Horner power1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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

After

g INNER at ORIGINAL x Horner power0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [g INNER at ORIGINAL x, [3, 1, 3, 1], -1]

After

derived = [4]

Conditions: none

Step identifiers and dependencies

Rule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

f derivative at INNER g(x) Horner power1 plus coefficient := sum(12, 4)

After

f derivative at INNER g(x) Horner power1 plus coefficient = 16

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

f derivative at INNER g(x) Horner power0 times argument := mul(16, 4)

After

f derivative at INNER g(x) Horner power0 times argument = 64

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

f derivative at INNER g(x) Horner power0 plus coefficient := sum(64, 1)

After

f derivative at INNER g(x) Horner power0 plus coefficient = 65

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [f derivative at INNER g(x), [1, 4, 3], 4]

After

derived = [65]

Conditions: none

Step identifiers and dependencies

Rule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S033

S035 · calculation

Before

Derivative chain ORIGINAL outer derivative at gx times inner derivative := mul(65, -2)

After

Derivative chain ORIGINAL outer derivative at gx times inner derivative = -130

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

source roles = [[-4, 1, 2, 1], [3, 1, 3, 1], -1, [1, 4, 3], [1, 6, 3]]

After

derived = [4, 65, -2, -130]

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 dependencies

Rule: 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

-130

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: 53e8c3d1b83bbf8f6b8e96a8a17c7c5ff848a1c84a64493ef302f09501ff3459
candidate record SHA-256: 735618c22724b39ab99ec5592949c3784ed60f1039c61afc1c740338153452cc
semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69
question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495
Lean: NOT_RUN · human review: NOT_PERFORMED

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