Question 10151: Derivative of an Oriented Integral with Moving Bounds

Original question and givens

On all R, continuous f has ascending coefficients [1], constant term first. Let lower l(x)=(1)x+(0) and upper u(x)=(2)x+(0). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=0. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, continuous f has ascending coefficients [1], constant term first. Let lower l(x)=(1)x+(0) and upper u(x)=(2)x+(0). Find the derivative of ∫_l(x)^u(x) f(t)dt at x=0. Use reduced positive-denominator fractions, retaining /1 for integers.

Definition and scope

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.

• mko-fundamental-theorem-calculus

Step-by-step derivation

raw integrand f=[1];ascending powers on all R

normalized coefficients=[1/1];declared continuous real polynomial

zero primitive coefficient1 := div(1, 1)

zero primitive coefficient1 = 1

integrand power=0;coefficient=1/1

primitive power=1;coefficient=1/1

original integrand coefficients=[1/1]

zero-constant primitive coefficients=[0/1, 1/1]

constructed P0 derivative coefficient0 := mul(1, 1)

constructed P0 derivative coefficient0 = 1

constructed P0:source power=1;coefficient=1/1

derivative power=0;coefficient=1/1

P0 derivative coefficients=[1/1];f=[1/1]

P0'=f on all R in this declared polynomial model

lower bound m*x := mul(1, 0)

lower bound m*x = 0

lower bound m*x+b := sum(0, 0)

lower bound m*x+b = 0

upper bound m*x := mul(2, 0)

upper bound m*x = 0

upper bound m*x+b := sum(0, 0)

upper bound m*x+b = 0

H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=1;upper slope=2

H'(x)=f(u(x))*upper_slope-f(l(x))*lower_slope

f at upper Horner multiply0 := mul(0, 0)

f at upper Horner multiply0 = 0

f at upper Horner add0 := sum(0, 1)

f at upper Horner add0 = 1

f at lower Horner multiply0 := mul(0, 0)

f at lower Horner multiply0 = 0

f at lower Horner add0 := sum(0, 1)

f at lower Horner add0 = 1

upper chain contribution := mul(1, 2)

upper chain contribution = 2

lower chain contribution := mul(1, 1)

lower chain contribution = 1

moving bound derivative upper-minus-lower := sub(2, 1)

moving bound derivative upper-minus-lower = 1

point=0;lower=0;upper=0;f(upper)=1;f(lower)=1

upper contribution=2;lower contribution=1;derivative=1/1

Reverse checks

• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute lower bound m*x 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute constructed P0 derivative coefficient0 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 zero primitive coefficient1 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.

Result

1/1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.