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R0006-03690 · q-20261007-advance-20261009-r04-derivative-rules-0090
On all R, f has ascending coefficients [1, 14, 16, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.
G001 — On all R, f has ascending coefficients [1, 14, 16, 3], constant term first and g has ascending coefficients [-1, 1, 1, 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=[1, 14, 16, 3]
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, 14)
After
f′ coefficient at degree0 = 14
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, 16)
After
f′ coefficient at degree1 = 32
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, 3)
After
f′ coefficient at degree2 = 9
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
sum derivative coefficient0 := sum(14, 1)
After
sum derivative coefficient0 = 15
Conditions: none
Add derivative coefficients of the same degree,using0 for a missing high-degree term.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
sum derivative coefficient1 := sum(32, 2)
After
sum derivative coefficient1 = 34
Conditions: none
Add derivative coefficients of the same degree,using0 for a missing high-degree term.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
sum derivative coefficient2 := sum(9, 3)
After
sum derivative coefficient2 = 12
Conditions: none
Add derivative coefficients of the same degree,using0 for a missing high-degree term.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
[15, 34, 12]
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, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute sum derivative coefficient2 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute sum derivative coefficient1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute sum derivative coefficient0 with exact rational arithmetic, not rounded decimals.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.[15, 34, 12]
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: b1cbc7056dfb41c619e1c170662a54a7d99cd397fe2cb66dcb02eecb3af30a8ecandidate record SHA-256: 9e40385cf2b97c1447803c23f8f3447129a61647268419a5ee9ee81913de9e28semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED