# Question 10101: Point Derivative of a Continuous Accumulating Integral [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-0001) 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. R0019-10101 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0001 ## Original question and givens On all R, the polynomial f is continuous and f has ascending coefficients [0], constant term first. Let G(x)=∫_-1^x f(t)dt with oriented endpoints. Find G′(-2). Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, the polynomial f is continuous and f has ascending coefficients [0], constant term first. Let G(x)=∫_-1^x f(t)dt with oriented endpoints. Find G′(-2). 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=[0];ascending powers on all R After normalized coefficients=[0/1];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(0, 1) After zero primitive coefficient1 = 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, S001 S003 · impliesBefore integrand power=0;coefficient=0/1 After primitive power=1;coefficient=0/1 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 · impliesBefore original integrand coefficients=[0/1] After zero-constant primitive coefficients=[0/1] 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, S003 S005 · impliesBefore constructed P0:constant coefficient=0/1 After derivative coefficients=[0/1] Conditions: none A constant differentiates to the zero polynomial,represented as[0/1],not a fabricated negative power. Step identifiers and dependenciesRule: constant_polynomial_derivative_zero · Uses: G001, S004 S006 · impliesBefore P0 derivative coefficients=[0/1];f=[0/1] 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, S005 S007 · calculationBefore P0 at anchor Horner multiply0 := mul(0, -1) After P0 at anchor Horner multiply0 = 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, S006 S008 · calculationBefore P0 at anchor Horner add0 := sum(0, 0) After P0 at anchor Horner add0 = 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, S007 S009 · impliesBefore anchor a=-1;P0(a)=0/1 After G(t)=P0(t)-(0/1);G'(t)=f(t) Conditions: none The anchor contributes the t-independent constant P0(a),which vanishes under differentiation.Do not use G's zero anchor value as its derivative. Step identifiers and dependenciesRule: FTC_accumulation_anchor_constant · Uses: G001, S008 S010 · calculationBefore f at derivative point Horner multiply0 := mul(0, -2) After f at derivative point Horner multiply0 = 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, S009 S011 · calculationBefore f at derivative point Horner add0 := sum(0, 0) After f at derivative point Horner add0 = 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, S010 ## Reverse checks - S011 · premise_and_result_audit — Recompute f at derivative point Horner add0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute f at derivative point Horner multiply0 with exact rational arithmetic, not rounded decimals. - 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 P0 at anchor Horner add0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute P0 at anchor Horner multiply0 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 0/1 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: 090ca8f2f025d9eaecbccc0363be2c9bf994017021b1e56c21c6a6b5362df514 - candidate record SHA-256: 789cba392868fe9b1c3efe0b571c666bc7100c5241386719e5dbe772e9989751 - 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-0001.json)