Question 33573: A Fixed-x Slice and Its y-Derivative

Original question and givens

Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] 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=0, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

• G001 — Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] 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=0, 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

Step-by-step derivation

source roles = [qf-mod25-partialdeep-partial-y-slice-derivative, [coefficients, domain, fixed_x, operation, y]]

derived = [partialdeep-partial-y-slice-derivative, Polynomial_partial_derivative_density]

source roles = [[10, 0, -2, 3, 0, 0]]

derived = [1, x, y, x^2, x*y, y^2]

Slice ORIGINAL c_x times fixed x := mul(0, 0)

Slice ORIGINAL c_x times fixed x = 0

Slice ORIGINAL fixed x squared := mul(0, 0)

Slice ORIGINAL fixed x squared = 0

Slice ORIGINAL c_xx times fixed x squared := mul(3, 0)

Slice ORIGINAL c_xx times fixed x squared = 0

Slice ORIGINAL constant terms sum := sum(10, 0, 0)

Slice ORIGINAL constant terms sum = 10

Slice ORIGINAL c_xy times fixed x := mul(0, 0)

Slice ORIGINAL c_xy times fixed x = 0

Slice ORIGINAL linear y terms sum := sum(-2, 0)

Slice ORIGINAL linear y terms sum = -2

Slice ORIGINAL twice c_yy := mul(2, 0)

Slice ORIGINAL twice c_yy = 0

source roles = [[10, 0, -2, 3, 0, 0], 0]

derived = [[10, -2, 0], [-2, 0]]

Slice ORIGINAL derivative linear term at y := mul(0, -2)

Slice ORIGINAL derivative linear term at y = 0

Slice ORIGINAL full derivative value := sum(-2, 0)

Slice ORIGINAL full derivative value = -2

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

([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 0], -2, -2)

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.