# Question 10165: Derivative of an Oriented Integral with Moving Bounds [Return to catalogue page 51](https://ocme.evemisslab.com/en/questions/page/51/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r01-fundamental-theorem-calculus-foundations/#q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0040) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0019-10165 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0040 ## Original question and givens On all R, continuous f has ascending coefficients [2, 15, 1, 3], constant term first. Let lower l(x)=(3)x+(-2) and upper u(x)=(6)x+(2). 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 [2, 15, 1, 3], constant term first. Let lower l(x)=(3)x+(-2) and upper u(x)=(6)x+(2). 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](https://ocme.evemisslab.com/en/mko/mko-fundamental-theorem-calculus/) ## Step-by-step derivation S001 · impliesBefore raw integrand f=[2, 15, 1, 3];ascending powers on all R After normalized coefficients=[2/1, 15/1, 1/1, 3/1];declared continuous real polynomial Conditions: none The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof. Step identifiers and dependenciesRule: declared_FTC_continuity_hypothesis · Uses: G001 S002 · calculationBefore zero primitive coefficient1 := div(2, 1) After zero primitive coefficient1 = 2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore integrand power=0;coefficient=2/1 After primitive power=1;coefficient=2/1 Conditions: none For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive. Step identifiers and dependenciesRule: polynomial_primitive_coefficient_rule · Uses: G001, S002 S004 · calculationBefore zero primitive coefficient2 := div(15, 2) After zero primitive coefficient2 = 15/2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore integrand power=1;coefficient=15/1 After primitive power=2;coefficient=15/2 Conditions: none For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive. Step identifiers and dependenciesRule: polynomial_primitive_coefficient_rule · Uses: G001, S004 S006 · calculationBefore zero primitive coefficient3 := div(1, 3) After zero primitive coefficient3 = 1/3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore integrand power=2;coefficient=1/1 After primitive power=3;coefficient=1/3 Conditions: none For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive. Step identifiers and dependenciesRule: polynomial_primitive_coefficient_rule · Uses: G001, S006 S008 · calculationBefore zero primitive coefficient4 := div(3, 4) After zero primitive coefficient4 = 3/4 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore integrand power=3;coefficient=3/1 After primitive power=4;coefficient=3/4 Conditions: none For c*x^k place c/(k+1) at x^(k+1),with positive denominator k+1.Differentiating recovers c*x^k.Any additive constant gives another primitive. Step identifiers and dependenciesRule: polynomial_primitive_coefficient_rule · Uses: G001, S008 S010 · impliesBefore original integrand coefficients=[2/1, 15/1, 1/1, 3/1] After zero-constant primitive coefficients=[0/1, 2/1, 15/2, 1/3, 3/4] Conditions: none This is one convenient representative,not a claim that the primitive is unique without its constant family. Step identifiers and dependenciesRule: zero_constant_polynomial_primitive · Uses: G001, S009 S011 · calculationBefore constructed P0 derivative coefficient0 := mul(1, 2) After constructed P0 derivative coefficient0 = 2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore constructed P0:source power=1;coefficient=2/1 After derivative power=0;coefficient=2/1 Conditions: none Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term. Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S011 S013 · calculationBefore constructed P0 derivative coefficient1 := mul(2, 15/2) After constructed P0 derivative coefficient1 = 15 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore constructed P0:source power=2;coefficient=15/2 After derivative power=1;coefficient=15/1 Conditions: none Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term. Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S013 S015 · calculationBefore constructed P0 derivative coefficient2 := mul(3, 1/3) After constructed P0 derivative coefficient2 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore constructed P0:source power=3;coefficient=1/3 After derivative power=2;coefficient=1/1 Conditions: none Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term. Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S015 S017 · calculationBefore constructed P0 derivative coefficient3 := mul(4, 3/4) After constructed P0 derivative coefficient3 = 3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore constructed P0:source power=4;coefficient=3/4 After derivative power=3;coefficient=3/1 Conditions: none Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term. Step identifiers and dependenciesRule: polynomial_derivative_coefficient_rule · Uses: G001, S017 S019 · impliesBefore P0 derivative coefficients=[2/1, 15/1, 1/1, 3/1];f=[2/1, 15/1, 1/1, 3/1] After P0'=f on all R in this declared polynomial model Conditions: none The full coefficient identity establishes the primitive relation in the declared model.With the supplied continuity premise use the FTC endpoint difference,not a function guessed from sample points. Step identifiers and dependenciesRule: constructed_primitive_coefficient_identity · Uses: G001, S018 S020 · calculationBefore lower bound m*x := mul(3, 0) After lower bound m*x = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore lower bound m*x+b := sum(0, -2) After lower bound m*x+b = -2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore upper bound m*x := mul(6, 0) After upper bound m*x = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore upper bound m*x+b := sum(0, 2) After upper bound m*x+b = 2 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore H(x)=P0(u(x))-P0(l(x));P0'=f;lower slope=3;upper slope=6 After H'(x)=f(u(x))*upper_slope-f(l(x))*lower_slope Conditions: none Use the oriented endpoint difference,then the chain rule for both compositions.Subtract the whole lower-bound term;constant bounds have slope0.Equal bound values at one point do not mean equal velocities or force a zero derivative. Step identifiers and dependenciesRule: FTC_moving_bounds_two_chain_factors · Uses: G001, S023 S025 · calculationBefore f at upper Horner multiply3 := mul(0, 2) After f at upper Horner multiply3 = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore f at upper Horner add3 := sum(0, 3) After f at upper Horner add3 = 3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore f at upper Horner multiply2 := mul(3, 2) After f at upper Horner multiply2 = 6 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore f at upper Horner add2 := sum(6, 1) After f at upper Horner add2 = 7 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore f at upper Horner multiply1 := mul(7, 2) After f at upper Horner multiply1 = 14 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore f at upper Horner add1 := sum(14, 15) After f at upper Horner add1 = 29 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore f at upper Horner multiply0 := mul(29, 2) After f at upper Horner multiply0 = 58 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore f at upper Horner add0 := sum(58, 2) After f at upper Horner add0 = 60 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore f at lower Horner multiply3 := mul(0, -2) After f at lower Horner multiply3 = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore f at lower Horner add3 := sum(0, 3) After f at lower Horner add3 = 3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore f at lower Horner multiply2 := mul(3, -2) After f at lower Horner multiply2 = -6 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore f at lower Horner add2 := sum(-6, 1) After f at lower Horner add2 = -5 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore f at lower Horner multiply1 := mul(-5, -2) After f at lower Horner multiply1 = 10 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore f at lower Horner add1 := sum(10, 15) After f at lower Horner add1 = 25 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore f at lower Horner multiply0 := mul(25, -2) After f at lower Horner multiply0 = -50 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore f at lower Horner add0 := sum(-50, 2) After f at lower Horner add0 = -48 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore upper chain contribution := mul(60, 6) After upper chain contribution = 360 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore lower chain contribution := mul(-48, 3) After lower chain contribution = -144 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore moving bound derivative upper-minus-lower := sub(360, -144) After moving bound derivative upper-minus-lower = 504 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore point=0;lower=-2;upper=2;f(upper)=60;f(lower)=-48 After upper contribution=360;lower contribution=-144;derivative=504/1 Conditions: none Retain both actual endpoints,integrand values and slopes,then subtract with sign.Do not take an absolute area or discard zero/negative velocities. Step identifiers and dependenciesRule: FTC_moving_bounds_actual_values · Uses: G001, S043 ## Reverse checks - 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 moving bound derivative upper-minus-lower with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute lower chain contribution with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute upper chain contribution with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute f at lower Horner add0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute f at lower Horner multiply0 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute f at lower Horner add1 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute f at lower Horner multiply1 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute f at lower Horner add2 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute f at lower Horner multiply2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute f at lower Horner add3 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute f at lower Horner multiply3 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute f at upper Horner add0 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute f at upper Horner multiply0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute f at upper Horner add1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute f at upper Horner multiply1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute f at upper Horner add2 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute f at upper Horner multiply2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute f at upper Horner add3 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute f at upper Horner multiply3 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute upper bound m*x+b with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute upper bound m*x with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute lower bound m*x+b with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute lower bound m*x with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute constructed P0 derivative coefficient3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute constructed P0 derivative coefficient2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute constructed P0 derivative coefficient1 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 constructed P0 derivative coefficient0 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 — Recompute zero primitive coefficient4 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 zero primitive coefficient3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute zero primitive coefficient2 with exact rational arithmetic, not rounded decimals. - 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 504/1 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a644043ac7e94071c2ba12a6e90187381eafd1fd3cad59027a76dfe8b678dfe3 - candidate record SHA-256: dc844effb7441ecedb92e158065c7d8bf724f1d19e3a17ede7a5f5d555302a5e - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0040.json)