Explore related knowledge and questions →
← Question derivation index · Original question batch
R0006-03611 · q-20261007-advance-20261009-r04-derivative-rules-0061
On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first and g has ascending coefficients [5, 1, 2, 1], constant term first. At x=1, 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 [-2, 11, 0, 3], constant term first and g has ascending coefficients [5, 1, 2, 1], constant term first. At x=1, 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=[-2, 11, 0, 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, 11)
After
f′ coefficient at degree0 = 11
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, 0)
After
f′ coefficient at degree1 = 0
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, 2)
After
g′ coefficient at degree1 = 4
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
f(x) ascending coefficients=[-2, 11, 0, 3]
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, S007
Before
f(x) Horner multiply before degree2 := mul(3, 1)
After
f(x) Horner multiply before degree2 = 3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
f(x) Horner add degree2 := sum(3, 0)
After
f(x) Horner add degree2 = 3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
f(x) Horner multiply before degree1 := mul(3, 1)
After
f(x) Horner multiply before degree1 = 3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
f(x) Horner add degree1 := sum(3, 11)
After
f(x) Horner add degree1 = 14
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
f(x) Horner multiply before degree0 := mul(14, 1)
After
f(x) Horner multiply before degree0 = 14
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
f(x) Horner add degree0 := sum(14, -2)
After
f(x) Horner add degree0 = 12
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
g(x) ascending coefficients=[5, 1, 2, 1]
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, S014
Before
g(x) Horner multiply before degree2 := mul(1, 1)
After
g(x) Horner multiply before degree2 = 1
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
g(x) Horner add degree2 := sum(1, 2)
After
g(x) Horner add degree2 = 3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
g(x) Horner multiply before degree1 := mul(3, 1)
After
g(x) Horner multiply before degree1 = 3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
g(x) Horner add degree1 := sum(3, 1)
After
g(x) Horner add degree1 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
g(x) Horner multiply before degree0 := mul(4, 1)
After
g(x) Horner multiply before degree0 = 4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
g(x) Horner add degree0 := sum(4, 5)
After
g(x) Horner add degree0 = 9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
f′(x) ascending coefficients=[11, 0, 9]
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, S021
Before
f′(x) Horner multiply before degree1 := mul(9, 1)
After
f′(x) Horner multiply before degree1 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
f′(x) Horner add degree1 := sum(9, 0)
After
f′(x) Horner add degree1 = 9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
f′(x) Horner multiply before degree0 := mul(9, 1)
After
f′(x) Horner multiply before degree0 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
f′(x) Horner add degree0 := sum(9, 11)
After
f′(x) Horner add degree0 = 20
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
g′(x) ascending coefficients=[1, 4, 3]
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, S026
Before
g′(x) Horner multiply before degree1 := mul(3, 1)
After
g′(x) Horner multiply before degree1 = 3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
g′(x) Horner add degree1 := sum(3, 4)
After
g′(x) Horner add degree1 = 7
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
g′(x) Horner multiply before degree0 := mul(7, 1)
After
g′(x) Horner multiply before degree0 = 7
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
g′(x) Horner add degree0 := sum(7, 1)
After
g′(x) Horner add degree0 = 8
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
f′(g(x)) ascending coefficients=[11, 0, 9]
After
evaluate at x=9
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, S031
Before
f′(g(x)) Horner multiply before degree1 := mul(9, 9)
After
f′(g(x)) Horner multiply before degree1 = 81
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
f′(g(x)) Horner add degree1 := sum(81, 0)
After
f′(g(x)) Horner add degree1 = 81
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
f′(g(x)) Horner multiply before degree0 := mul(81, 9)
After
f′(g(x)) Horner multiply before degree0 = 729
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
f′(g(x)) Horner add degree0 := sum(729, 11)
After
f′(g(x)) Horner add degree0 = 740
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
g′(f(x)) ascending coefficients=[1, 4, 3]
After
evaluate at x=12
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, S036
Before
g′(f(x)) Horner multiply before degree1 := mul(3, 12)
After
g′(f(x)) Horner multiply before degree1 = 36
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
g′(f(x)) Horner add degree1 := sum(36, 4)
After
g′(f(x)) Horner add degree1 = 40
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
g′(f(x)) Horner multiply before degree0 := mul(40, 12)
After
g′(f(x)) Horner multiply before degree0 = 480
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
g′(f(x)) Horner add degree0 := sum(480, 1)
After
g′(f(x)) Horner add degree0 = 481
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
(f∘g)′(x) := mul(740, 8)
After
(f∘g)′(x) = 5920
Conditions: none
Evaluate the outer derivative f′ at g(x),then multiply by the inner derivative g′(x).
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
(g∘f)′(x) := mul(481, 20)
After
(g∘f)′(x) = 9620
Conditions: none
The reverse order changes the outer function,the evaluation point and the inner derivative.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
inner values=(9,12);inner derivatives=(8,20)
After
ordered derivative values=(5920/1, 9620/1)
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, S043
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Recompute (g∘f)′(x) with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute (f∘g)′(x) with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute g′(f(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute g′(f(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree0 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute f′(g(x)) Horner add degree1 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute f′(g(x)) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Recompute g′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute g′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute g′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Recompute f′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute f′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute f′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute g(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute g(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute g(x) Horner add degree2 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute g(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute f(x) Horner add degree0 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute f(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute f(x) Horner add degree1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute f(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute f(x) Horner add degree2 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute f(x) Horner multiply before degree2 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 — 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.(5920/1, 9620/1)
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: f6490badcfafc9415e6e7ab509019e52af169620a6a912bdf638ef78d3228683candidate record SHA-256: c6106dd5b42c6780b2b4f65bda8edbf6f7da13c17ca03f597b45ec42f3d79d4esemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED