# Question 33570: A Fixed-x Slice and Its y-Derivative [Return to catalogue page 168](https://ocme.evemisslab.com/en/questions/page/168/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r07-partial-derivative-foundations/#q-20261013-advance-20261025-r07-partial-derivative-0070) 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. R0066-33570 · q-20261013-advance-20261025-r07-partial-derivative-0070 ## Original question and givens Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value). - G001 — Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value). ## 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-partial-derivative](https://ocme.evemisslab.com/en/mko/mko-partial-derivative/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod25-partialdeep-partial-y-slice-derivative, [coefficients, domain, fixed_x, operation, y]] After derived = [partialdeep-partial-y-slice-derivative, Polynomial_partial_derivative_density] Conditions: none THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged. Step identifiers and dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001 S002 · impliesBefore source roles = [[7, 2, 2, 4, 0, 2]] After derived = [1, x, y, x^2, x*y, y^2] Conditions: none The SIX coefficients use the full[1,x,y,x²,xy,y²]basis.Hold the OTHER coordinate fixed.This supplied polynomial supports symbolic identities,not inference of a derivative from one difference sample or total differentiability of arbitrary functions. Step identifiers and dependenciesRule: Partial_FULL_polynomial_basis_and_fixed_other_coordinate · Uses: G001, S001 S003 · calculationBefore Slice ORIGINAL c_x times fixed x := mul(2, 2) After Slice ORIGINAL c_x times fixed x = 4 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Slice ORIGINAL fixed x squared := mul(2, 2) After Slice ORIGINAL fixed x squared = 4 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Slice ORIGINAL c_xx times fixed x squared := mul(4, 4) After Slice ORIGINAL c_xx times fixed x squared = 16 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Slice ORIGINAL constant terms sum := sum(7, 4, 16) After Slice ORIGINAL constant terms sum = 27 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Slice ORIGINAL c_xy times fixed x := mul(0, 2) After Slice ORIGINAL c_xy times fixed x = 0 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Slice ORIGINAL linear y terms sum := sum(2, 0) After Slice ORIGINAL linear y terms sum = 2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Slice ORIGINAL twice c_yy := mul(2, 2) After Slice ORIGINAL twice c_yy = 4 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [[7, 2, 2, 4, 0, 2], 2] After derived = [[27, 2, 2], [2, 4]] Conditions: none FIRST fix x and collect the WHOLE y-polynomial s0+s1*y+s2*y².THEN differentiate to s1+2*s2*y.Do not substitute BOTH coordinates into a constant before differentiating. Step identifiers and dependenciesRule: Partial_FIXED_x_whole_y_polynomial_THEN_derivative · Uses: G001, S009 S011 · calculationBefore Slice ORIGINAL derivative linear term at y := mul(4, 2) After Slice ORIGINAL derivative linear term at y = 8 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Slice ORIGINAL full derivative value := sum(2, 8) After Slice ORIGINAL full derivative value = 10 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Recompute Slice ORIGINAL full derivative value with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Slice ORIGINAL derivative linear term at y with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S009 · premise_and_result_audit — Recompute Slice ORIGINAL twice c_yy with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Slice ORIGINAL linear y terms sum with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Slice ORIGINAL c_xy times fixed x with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Slice ORIGINAL constant terms sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Slice ORIGINAL c_xx times fixed x squared with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Slice ORIGINAL fixed x squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Slice ORIGINAL c_x times fixed x with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. ## Result ([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10) 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: 8871886eae12d10a4eb1c34b62e176bc4415c844d791e8b5b44bef1365ad8262 - candidate record SHA-256: 0a6404136222a552ec2f75fb3fb779a6f3df849125213b587e11a6fd54f683d8 - semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267 - question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r07-partial-derivative-0070.json)