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R0006-03608 · q-20261007-advance-20261009-r04-derivative-rules-0058
On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, 1, 2, 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 [-5, 8, 0, 4], constant term first and g has ascending coefficients [2, 1, 2, 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=[-5, 8, 0, 4]
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, 8)
After
f′ coefficient at degree0 = 8
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, 4)
After
f′ coefficient at degree2 = 12
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=[-5, 8, 0, 4]
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(x) Horner multiply before degree2 := mul(4, 2)
After
f(x) Horner multiply before degree2 = 8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
f(x) Horner add degree2 := sum(8, 0)
After
f(x) Horner add degree2 = 8
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(8, 2)
After
f(x) Horner multiply before degree1 = 16
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
f(x) Horner add degree1 := sum(16, 8)
After
f(x) Horner add degree1 = 24
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(24, 2)
After
f(x) Horner multiply before degree0 = 48
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
f(x) Horner add degree0 := sum(48, -5)
After
f(x) Horner add degree0 = 43
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
g(x) ascending coefficients=[2, 1, 2, 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, S014
Before
g(x) Horner multiply before degree2 := mul(1, 2)
After
g(x) Horner multiply before degree2 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
g(x) Horner add degree2 := sum(2, 2)
After
g(x) Horner add degree2 = 4
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(4, 2)
After
g(x) Horner multiply before degree1 = 8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
g(x) Horner add degree1 := sum(8, 1)
After
g(x) Horner add degree1 = 9
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(9, 2)
After
g(x) Horner multiply before degree0 = 18
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
g(x) Horner add degree0 := sum(18, 2)
After
g(x) Horner add degree0 = 20
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
f′(x) ascending coefficients=[8, 0, 12]
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, S021
Before
f′(x) Horner multiply before degree1 := mul(12, 2)
After
f′(x) Horner multiply before degree1 = 24
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
f′(x) Horner add degree1 := sum(24, 0)
After
f′(x) Horner add degree1 = 24
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(24, 2)
After
f′(x) Horner multiply before degree0 = 48
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
f′(x) Horner add degree0 := sum(48, 8)
After
f′(x) Horner add degree0 = 56
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=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, S026
Before
g′(x) Horner multiply before degree1 := mul(3, 2)
After
g′(x) Horner multiply before degree1 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
g′(x) Horner add degree1 := sum(6, 4)
After
g′(x) Horner add degree1 = 10
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(10, 2)
After
g′(x) Horner multiply before degree0 = 20
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
g′(x) Horner add degree0 := sum(20, 1)
After
g′(x) Horner add degree0 = 21
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
f′(g(x)) ascending coefficients=[8, 0, 12]
After
evaluate at x=20
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(12, 20)
After
f′(g(x)) Horner multiply before degree1 = 240
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
f′(g(x)) Horner add degree1 := sum(240, 0)
After
f′(g(x)) Horner add degree1 = 240
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(240, 20)
After
f′(g(x)) Horner multiply before degree0 = 4800
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
f′(g(x)) Horner add degree0 := sum(4800, 8)
After
f′(g(x)) Horner add degree0 = 4808
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=43
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, 43)
After
g′(f(x)) Horner multiply before degree1 = 129
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
g′(f(x)) Horner add degree1 := sum(129, 4)
After
g′(f(x)) Horner add degree1 = 133
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(133, 43)
After
g′(f(x)) Horner multiply before degree0 = 5719
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
g′(f(x)) Horner add degree0 := sum(5719, 1)
After
g′(f(x)) Horner add degree0 = 5720
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
(f∘g)′(x) := mul(4808, 21)
After
(f∘g)′(x) = 100968
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(5720, 56)
After
(g∘f)′(x) = 320320
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=(20,43);inner derivatives=(21,56)
After
ordered derivative values=(100968/1, 320320/1)
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.(100968/1, 320320/1)
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.
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