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R0006-03676 · q-20261007-advance-20261009-r04-derivative-rules-0076
On all R, f has ascending coefficients [2, 1], constant term first and g has ascending coefficients [-4, -1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.
G001 — On all R, f has ascending coefficients [2, 1], constant term first and g has ascending coefficients [-4, -1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.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=[2, 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.
Rule: polynomial_derivative_premise · Uses: G001
Before
f′ coefficient at degree0 := mul(1, 1)
After
f′ 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
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, S002
Before
sum derivative coefficient0 := sum(1, -1)
After
sum derivative coefficient0 = 0
Conditions: none
Add derivative coefficients of the same degree,using0 for a missing high-degree term.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
[0]
After
complete derivative coefficient identity;zero polynomial=[0]
Conditions: none
The sum rule gives(f+g)′=f′+g′.Removing trailing zeros preserves the polynomial;retain[0] for zero.This is not coefficient inference from a few sample points.
Rule: sum_rule_coefficient_identity · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute sum derivative coefficient0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute g′ coefficient at degree0 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.[0]
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: f114bf06ac9568bc91c59d5ae6839cdbdc8617b2bcdead2be68e8d8364aee1bbcandidate record SHA-256: d52a2763fcaeb68a60271cb0250481d0fc73ef06dccafe70bfac0c17da52d0f9semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED