Use the quadratic candidate coefficients=[4, -1, -1, 1, 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=-1; 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=[4, -1, -1, 1, 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=-1; 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.
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.
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.
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.
Quotient ORIGINAL twice c_xx times x := mul(2, -1)
After
Quotient ORIGINAL twice c_xx times x = -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.
Quotient ORIGINAL c_xy times fixed y := mul(0, -1)
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.
Quotient ORIGINAL all linear h coefficients := sum(-1, -2, 0)
After
Quotient ORIGINAL all linear h coefficients = -3
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.
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.
derived = [[-3, 1], -3, 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.
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.
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.