Question 11594: A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Original question and givens

Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.

• G001 — Use coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. A partial derivative fixes the other coordinate and is defined by the one-variable difference-quotient limit as nonzero h tends to 0; the exercise uses exact polynomial differentiation, not a single finite-difference sample. Existence of partial derivatives alone establishes no general continuity, total differentiability or exchange of mixed derivatives. fixed y=1, evaluation x=1. Give (original coefficients,fixed y,ascending x-slice coefficients,ascending derivative coefficients,x,derivative value). The slice has degrees up to 2 and its derivative degrees up to 1.

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

original coefficients=[6, 1, 1, 1, -1, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(6)+(1)*x+(1)*y+(1)*x^2+(-1)*x*y+(-1)*y^2

fixed-y slice fixed coordinate squared := mul(1, 1)

fixed-y slice fixed coordinate squared = 1

fixed-y slice fixed linear term := mul(1, 1)

fixed-y slice fixed linear term = 1

fixed-y slice fixed square term := mul(-1, 1)

fixed-y slice fixed square term = -1

fixed-y slice constant coefficient := sum(6, 1, -1)

fixed-y slice constant coefficient = 6

fixed-y slice cross contribution := mul(-1, 1)

fixed-y slice cross contribution = -1

fixed-y slice linear coefficient := sum(1, -1)

fixed-y slice linear coefficient = 0

fixed-y slice:vary x;fix y=1;original coefficients=[6, 1, 1, 1, -1, -1]

ascending x slice coefficients=[6, 0, 1]

x slice derivative:g(t)=(6)+(0)*t+(1)*t^2;point=1;h!=0

g(1+h)-g(1)=(0)*h+(1)*(2*(1)*h+h^2)

x slice derivative twice quadratic coefficient := mul(2, 1)

x slice derivative twice quadratic coefficient = 2

x slice derivative derivative point term := mul(2, 1)

x slice derivative derivative point term = 2

x slice derivative derivative value := sum(0, 2)

x slice derivative derivative value = 2

x slice derivative:h!=0;increment=h*((0)+2*(1)*(1)+(1)*h)

difference quotient=2+(1)*h

x slice derivative:quotient=2+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(2))=abs(1)*abs(h)<epsilon;limit=2

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute x slice derivative derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute x slice derivative derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute x slice derivative twice quadratic coefficient 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute fixed-y slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute fixed-y slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute fixed-y slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute fixed-y slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute fixed-y slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute fixed-y slice fixed coordinate squared 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

([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 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.