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R0006-03657 · q-20261007-advance-20261009-r04-derivative-rules-0007
On all R, f has ascending coefficients [-6, 1, 7, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f has ascending coefficients [-6, 1, 7, -1], constant term first. Find f″(-1). Use reduced positive-denominator fractions, retaining /1 for integers.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=[-6, 1, 7, -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
f′ coefficient at degree1 := mul(2, 7)
After
f′ coefficient at degree1 = 14
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, -1)
After
f′ 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, S003
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, S004
Before
f″ coefficient at degree1 := mul(2, -3)
After
f″ coefficient at degree1 = -6
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
f″(x) ascending coefficients=[14, -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.
Rule: polynomial_evaluation_order · Uses: G001, S006
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
f″(x) Horner add degree0 := sum(6, 14)
After
f″(x) Horner add degree0 = 20
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
first coefficients=[1, 14, -3];second=[14, -6]
After
f″(-1)=20
Conditions: none
Differentiate twice using each current degree;constants and linear polynomials have second derivative0.Do not square the first derivative.
Rule: successive_polynomial_derivatives · Uses: G001, S009
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute f″(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute f″(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.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 f″ coefficient at degree1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute f″ 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.20/1
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: a070a166441c9b20d16fc468c7d37945709d9cca5ddacba59bfb939abd542234candidate record SHA-256: 365c0038a325c680ce5c9d6326ff9e440756899dd7d45265d27d412d334639c4semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED