# Question 33577: Cancel Only Nonzero h before Taking the Symbolic Slice Limit [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-0077) 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. R0066-33577 · q-20261013-advance-20261025-r07-partial-derivative-0077 ## Original question and givens Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] 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. point x=-1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference). - G001 — Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] 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. point x=-1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference). ## 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-slice-difference-quotient-limit, [coefficients, domain, operation, x, y]] After derived = [partialdeep-slice-difference-quotient-limit, 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 = [[-11, -1, -2, 2, 0, -1]] 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 Quotient ORIGINAL twice c_xx := mul(2, 2) After Quotient ORIGINAL twice c_xx = 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 Quotient ORIGINAL twice c_xx times x := mul(4, -1) After Quotient ORIGINAL twice c_xx times 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, S003 S005 · calculationBefore Quotient ORIGINAL c_xy times fixed y := mul(0, -2) After Quotient ORIGINAL c_xy times fixed y = 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, S004 S006 · calculationBefore Quotient ORIGINAL all linear h coefficients := sum(-1, -4, 0) After Quotient ORIGINAL all linear h coefficients = -5 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 · impliesBefore source roles = [[-11, -1, -2, 2, 0, -1], -1, -2] After derived = [u(x+h,y)-u(x,y)=(-5)*h+(2)*h^2] Conditions: none Expand and subtract ALL six original terms:constant/y/y² cancel,the x term gives c_x*h,the x² term gives2*c_xx*x*h+c_xx*h²,and xy gives c_xy*y*h.This is a symbolic identity for every h. Step identifiers and dependenciesRule: Partial_SYMBOLIC_difference_expand_ALL_six_terms · Uses: G001, S006 S008 · impliesBefore source roles = [-5, 2, -1, -2] After derived = [[-5, 2], -5, h!=0 before division;h->0 afterwards] Conditions: none Divide by h ONLY for h≠0,obtaining constant+c_xx*h.THEN polynomial continuity gives limit constant as h→0.The original quotient is undefined at h=0;never substitute0 into0/0.A finite difference is not this limit proof. Step identifiers and dependenciesRule: Partial_CANCEL_h_ONLY_nonzero_THEN_polynomial_limit · Uses: G001, S007 ## Reverse checks - S008 · 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. - S007 · 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. - S006 · premise_and_result_audit — Recompute Quotient ORIGINAL all linear h coefficients with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Quotient ORIGINAL c_xy times fixed y with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Quotient ORIGINAL twice c_xx times x with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Quotient ORIGINAL twice c_xx 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 ([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference) 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: 3b5043f8efc617f7a329e9d620095e9c7716dc6b6335207a690c27a885eba4e9 - candidate record SHA-256: 1ac42514cd0805bc77cd0e12758b93aada41cb3a66809592ef2ada052c886b48 - 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-0077.json)