Question 3605: Derivatives in Both Composition Orders

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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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R0006-03605 · q-20261007-advance-20261009-r04-derivative-rules-0055

Original question and givens

On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [-1, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

G001 — On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [-1, 1, 2, 1], constant term first. At x=-1, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.

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

f ascending coefficients=[-8, 5, 0, 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 dependencies

Rule: polynomial_derivative_premise · Uses: G001

S002 · calculation

Before

f′ coefficient at degree0 := mul(1, 5)

After

f′ coefficient at degree0 = 5

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

f′ coefficient at degree1 := mul(2, 0)

After

f′ coefficient at degree1 = 0

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

g′ coefficient at degree1 := mul(2, 2)

After

g′ coefficient at degree1 = 4

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

f(x) ascending coefficients=[-8, 5, 0, 1]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S007

S009 · calculation

Before

f(x) Horner multiply before degree2 := mul(1, -1)

After

f(x) Horner multiply before degree2 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

f(x) Horner add degree2 := sum(-1, 0)

After

f(x) Horner add degree2 = -1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

f(x) Horner multiply before degree1 := mul(-1, -1)

After

f(x) Horner multiply before degree1 = 1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

f(x) Horner add degree1 := sum(1, 5)

After

f(x) Horner add degree1 = 6

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

f(x) Horner multiply before degree0 := mul(6, -1)

After

f(x) Horner multiply before degree0 = -6

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

f(x) Horner add degree0 := sum(-6, -8)

After

f(x) Horner add degree0 = -14

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

g(x) ascending coefficients=[-1, 1, 2, 1]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S014

S016 · calculation

Before

g(x) Horner multiply before degree2 := mul(1, -1)

After

g(x) Horner multiply before degree2 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

g(x) Horner add degree2 := sum(-1, 2)

After

g(x) Horner add degree2 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

g(x) Horner multiply before degree1 := mul(1, -1)

After

g(x) Horner multiply before degree1 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

g(x) Horner add degree1 := sum(-1, 1)

After

g(x) Horner add degree1 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

g(x) Horner multiply before degree0 := mul(0, -1)

After

g(x) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

g(x) Horner add degree0 := sum(0, -1)

After

g(x) Horner add degree0 = -1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

f′(x) ascending coefficients=[5, 0, 3]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S021

S023 · calculation

Before

f′(x) Horner multiply before degree1 := mul(3, -1)

After

f′(x) Horner multiply before degree1 = -3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

f′(x) Horner add degree1 := sum(-3, 0)

After

f′(x) Horner add degree1 = -3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

f′(x) Horner multiply before degree0 := mul(-3, -1)

After

f′(x) Horner multiply before degree0 = 3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

f′(x) Horner add degree0 := sum(3, 5)

After

f′(x) Horner add degree0 = 8

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

g′(x) ascending coefficients=[1, 4, 3]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S026

S028 · calculation

Before

g′(x) Horner multiply before degree1 := mul(3, -1)

After

g′(x) Horner multiply before degree1 = -3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

g′(x) Horner add degree1 := sum(-3, 4)

After

g′(x) Horner add degree1 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

g′(x) Horner multiply before degree0 := mul(1, -1)

After

g′(x) Horner multiply before degree0 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

g′(x) Horner add degree0 := sum(-1, 1)

After

g′(x) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

f′(g(x)) ascending coefficients=[5, 0, 3]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S031

S033 · calculation

Before

f′(g(x)) Horner multiply before degree1 := mul(3, -1)

After

f′(g(x)) Horner multiply before degree1 = -3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

f′(g(x)) Horner add degree1 := sum(-3, 0)

After

f′(g(x)) Horner add degree1 = -3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

f′(g(x)) Horner multiply before degree0 := mul(-3, -1)

After

f′(g(x)) Horner multiply before degree0 = 3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

f′(g(x)) Horner add degree0 := sum(3, 5)

After

f′(g(x)) Horner add degree0 = 8

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

g′(f(x)) ascending coefficients=[1, 4, 3]

After

evaluate at x=-14

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S036

S038 · calculation

Before

g′(f(x)) Horner multiply before degree1 := mul(3, -14)

After

g′(f(x)) Horner multiply before degree1 = -42

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

g′(f(x)) Horner add degree1 := sum(-42, 4)

After

g′(f(x)) Horner add degree1 = -38

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

g′(f(x)) Horner multiply before degree0 := mul(-38, -14)

After

g′(f(x)) Horner multiply before degree0 = 532

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

g′(f(x)) Horner add degree0 := sum(532, 1)

After

g′(f(x)) Horner add degree0 = 533

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

(f∘g)′(x) := mul(8, 0)

After

(f∘g)′(x) = 0

Conditions: none

Evaluate the outer derivative f′ at g(x),then multiply by the inner derivative g′(x).

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

(g∘f)′(x) := mul(533, 8)

After

(g∘f)′(x) = 4264

Conditions: none

The reverse order changes the outer function,the evaluation point and the inner derivative.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

inner values=(-1,-14);inner derivatives=(0,8)

After

ordered derivative values=(0/1, 4264/1)

Conditions: none

The entries are f after g and g after f in that order;do not swap them.A constant outer function naturally makes its corresponding composite derivative0.

Step identifiers and dependencies

Rule: chain_rule_order_guard · Uses: G001, S043

Reverse checks

S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S043 · premise_and_result_audit — Recompute (g∘f)′(x) with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute (f∘g)′(x) with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S036 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Recompute g′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute g′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S026 · premise_and_result_audit — Recompute f′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute f′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute g(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute g(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute g(x) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute g(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Recompute f(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute f(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute f(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute f(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute f(x) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute f(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(0/1, 4264/1)

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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 8f2311264215be92b03a8563aa27be66cfbc7c5d52879387d71b499667ccbbae
candidate record SHA-256: 7a0ded59c40869b8f2f1ddc0cebe10f3c9998928ebdc9a61eb7dfc4ccec7dbeb
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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