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R0006-03626 · q-20261007-advance-20261009-r04-derivative-rules-0026
On all R, f has ascending coefficients [0], constant term first and g has ascending coefficients [2, 1], constant term first. At x=3, is 0 the derivative of fg?
G001 — On all R, f has ascending coefficients [0], constant term first and g has ascending coefficients [2, 1], constant term first. At x=3, is 0 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=[0]
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
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, S001
Before
f(x) ascending coefficients=[0]
After
evaluate at x=3
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, S002
Before
g(x) ascending coefficients=[2, 1]
After
evaluate at x=3
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, S003
Before
g(x) Horner multiply before degree0 := mul(1, 3)
After
g(x) Horner multiply before degree0 = 3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
g(x) Horner add degree0 := sum(3, 2)
After
g(x) Horner add degree0 = 5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
f′(x) ascending coefficients=[0]
After
evaluate at x=3
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, S006
Before
g′(x) ascending coefficients=[1]
After
evaluate at x=3
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)g(x) := mul(0, 5)
After
f′(x)g(x) = 0
Conditions: none
First product-rule term:differentiate f and retain g.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
f(x)g′(x) := mul(0, 1)
After
f(x)g′(x) = 0
Conditions: none
The second term differentiates only g;do not use f′g′.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
product derivative := sum(0, 0)
After
product derivative = 0
Conditions: none
Add both terms to obtain(fg)′(x).
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
candidate=0;actual product derivative=0
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, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute product derivative with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute f(x)g′(x) with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute f′(x)g(x) 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute g′ 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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: daedf5d898a8f2197ddd4b6f6751e47ce52af61b7f9c7ad3af9007f038508760candidate record SHA-256: 0bdb2f8c8e8cd1408978da6af26b8adddfb45bdfd815e8beb9da9687d145f49dsemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED