Question 3630: Eligibility of a Candidate Product-Derivative Value

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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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R0006-03630 · q-20261007-advance-20261009-r04-derivative-rules-0030

Original question and givens

On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -25 the derivative of fg?

G001 — On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first and g has ascending coefficients [2, -1, 1, 2], constant term first. At x=-1, is -25 the derivative of fg?

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, 1)

After

g′ coefficient at degree1 = 2

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, 2)

After

g′ coefficient at degree2 = 6

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=[2, -1, 1, 2]

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(2, -1)

After

g(x) Horner multiply before degree2 = -2

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(-2, 1)

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, 2)

After

g(x) Horner add degree0 = 2

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, 2, 6]

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(6, -1)

After

g′(x) Horner multiply before degree1 = -6

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(-6, 2)

After

g′(x) Horner add degree1 = -4

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(-4, -1)

After

g′(x) Horner multiply before degree0 = 4

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(4, -1)

After

g′(x) Horner add degree0 = 3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

f′(x)g(x) := mul(8, 2)

After

f′(x)g(x) = 16

Conditions: none

First product-rule term:differentiate f and retain g.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

f(x)g′(x) := mul(-14, 3)

After

f(x)g′(x) = -42

Conditions: none

The second term differentiates only g;do not use f′g′.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

product derivative := sum(16, -42)

After

product derivative = -26

Conditions: none

Add both terms to obtain(fg)′(x).

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

candidate=-25;actual product derivative=-26

After

No

Conditions: none

Compare the candidate at this specified x;the differentiation rule rests on polynomial differentiability,not a one-point coincidence.

Step identifiers and dependencies

Rule: product_rule_candidate_comparison · Uses: G001, S034

Reverse checks

S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S034 · premise_and_result_audit — Recompute product derivative with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute f(x)g′(x) with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute f′(x)g(x) with exact rational arithmetic, not rounded decimals.
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

No

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: bf14cc69f2ca41f2a3464a281bb7ef7110425f56ab5b75a1d754b383f9ec9cf3
candidate record SHA-256: cf54be2673235e028883536d56f7fdb0c02747f2466d89fd70ea4eb17b4e2f12
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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