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R0006-03641 · q-20261007-advance-20261009-r04-derivative-rules-0041
On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 1665 the derivative of fg?
G001 — On all R, f has ascending coefficients [3, 16, -1, 4], constant term first and g has ascending coefficients [2, -1, 1, 1], constant term first. At x=2, is 1665 the derivative of fg?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.
Before
f ascending coefficients=[3, 16, -1, 4]
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.
Rule: polynomial_derivative_premise · Uses: G001
Before
f′ coefficient at degree0 := mul(1, 16)
After
f′ coefficient at degree0 = 16
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
f′ coefficient at degree1 := mul(2, -1)
After
f′ coefficient at degree1 = -2
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
f′ coefficient at degree2 := mul(3, 4)
After
f′ coefficient at degree2 = 12
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S003
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
f(x) ascending coefficients=[3, 16, -1, 4]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S007
Before
f(x) Horner multiply before degree2 := mul(4, 2)
After
f(x) Horner multiply before degree2 = 8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
f(x) Horner add degree2 := sum(8, -1)
After
f(x) Horner add degree2 = 7
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
f(x) Horner multiply before degree1 := mul(7, 2)
After
f(x) Horner multiply before degree1 = 14
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
f(x) Horner add degree1 := sum(14, 16)
After
f(x) Horner add degree1 = 30
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
f(x) Horner multiply before degree0 := mul(30, 2)
After
f(x) Horner multiply before degree0 = 60
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
f(x) Horner add degree0 := sum(60, 3)
After
f(x) Horner add degree0 = 63
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
g(x) ascending coefficients=[2, -1, 1, 1]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S014
Before
g(x) Horner multiply before degree2 := mul(1, 2)
After
g(x) Horner multiply before degree2 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
g(x) Horner add degree2 := sum(2, 1)
After
g(x) Horner add degree2 = 3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
g(x) Horner multiply before degree1 := mul(3, 2)
After
g(x) Horner multiply before degree1 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
g(x) Horner add degree1 := sum(6, -1)
After
g(x) Horner add degree1 = 5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
g(x) Horner multiply before degree0 := mul(5, 2)
After
g(x) Horner multiply before degree0 = 10
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
g(x) Horner add degree0 := sum(10, 2)
After
g(x) Horner add degree0 = 12
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
f′(x) ascending coefficients=[16, -2, 12]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S021
Before
f′(x) Horner multiply before degree1 := mul(12, 2)
After
f′(x) Horner multiply before degree1 = 24
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
f′(x) Horner add degree1 := sum(24, -2)
After
f′(x) Horner add degree1 = 22
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
f′(x) Horner multiply before degree0 := mul(22, 2)
After
f′(x) Horner multiply before degree0 = 44
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
f′(x) Horner add degree0 := sum(44, 16)
After
f′(x) Horner add degree0 = 60
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
g′(x) ascending coefficients=[-1, 2, 3]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S026
Before
g′(x) Horner multiply before degree1 := mul(3, 2)
After
g′(x) Horner multiply before degree1 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
g′(x) Horner add degree1 := sum(6, 2)
After
g′(x) Horner add degree1 = 8
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
g′(x) Horner multiply before degree0 := mul(8, 2)
After
g′(x) Horner multiply before degree0 = 16
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
g′(x) Horner add degree0 := sum(16, -1)
After
g′(x) Horner add degree0 = 15
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
f′(x)g(x) := mul(60, 12)
After
f′(x)g(x) = 720
Conditions: none
First product-rule term:differentiate f and retain g.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
f(x)g′(x) := mul(63, 15)
After
f(x)g′(x) = 945
Conditions: none
The second term differentiates only g;do not use f′g′.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
product derivative := sum(720, 945)
After
product derivative = 1665
Conditions: none
Add both terms to obtain(fg)′(x).
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
candidate=1665;actual product derivative=1665
After
Yes
Conditions: none
Compare the candidate at this specified x;the differentiation rule rests on polynomial differentiability,not a one-point coincidence.
Rule: product_rule_candidate_comparison · Uses: G001, S034
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.Yes
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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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