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R0006-03601 · q-20261007-advance-20261009-r04-derivative-rules-0051
On all R, f has ascending coefficients [3], constant term first and g has ascending coefficients [1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f has ascending coefficients [3], constant term first and g has ascending coefficients [1, 1], constant term first. At x=2, give ((f∘g)′,(g∘f)′) in that order. 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=[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
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=[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, S002
Before
g(x) ascending coefficients=[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, S003
Before
g(x) Horner multiply before degree0 := mul(1, 2)
After
g(x) Horner multiply before degree0 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
g(x) Horner add degree0 := sum(2, 1)
After
g(x) Horner add degree0 = 3
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=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, S006
Before
g′(x) ascending coefficients=[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, S007
Before
f′(g(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, S008
Before
g′(f(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, S009
Before
(f∘g)′(x) := mul(0, 1)
After
(f∘g)′(x) = 0
Conditions: none
Evaluate the outer derivative f′ at g(x),then multiply by the inner derivative g′(x).
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
(g∘f)′(x) := mul(1, 0)
After
(g∘f)′(x) = 0
Conditions: none
The reverse order changes the outer function,the evaluation point and the inner derivative.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
inner values=(3,3);inner derivatives=(1,0)
After
ordered derivative values=(0/1, 0/1)
Conditions: none
The entries are f after g and g after f in that order;do not swap them.A constant outer function naturally makes its corresponding composite derivative0.
Rule: chain_rule_order_guard · Uses: G001, S012
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute (g∘f)′(x) with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute (f∘g)′(x) with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(0/1, 0/1)
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: 33a73407c3cd04826d750b831dcdf74a15fe0e4b8b1358fee268db4575aa7f09candidate record SHA-256: 697519cad34f66a1dda48c813fba9a6563920941f49d3d295380fcad4de2720fsemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED