Question 11542: Differentiate in the First Coordinate while Holding the Second Fixed

Original question and givens

Use coefficients=[4, -1, -1, 1, 0, -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. Evaluate at x=-1, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

• G001 — Use coefficients=[4, -1, -1, 1, 0, -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. Evaluate at x=-1, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

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=[4, -1, -1, 1, 0, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(4)+(-1)*x+(-1)*y+(1)*x^2+(0)*x*y+(-1)*y^2

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

x partial slice fixed coordinate squared = 1

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

x partial slice fixed linear term = 1

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

x partial slice fixed square term = -1

x partial slice constant coefficient := sum(4, 1, -1)

x partial slice constant coefficient = 4

x partial slice cross contribution := mul(0, -1)

x partial slice cross contribution = 0

x partial slice linear coefficient := sum(-1, 0)

x partial slice linear coefficient = -1

x partial slice:vary x;fix y=-1;original coefficients=[4, -1, -1, 1, 0, -1]

ascending x slice coefficients=[4, -1, 1]

x partial:g(t)=(4)+(-1)*t+(1)*t^2;point=-1;h!=0

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

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

x partial twice quadratic coefficient = 2

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

x partial derivative point term = -2

x partial derivative value := sum(-1, -2)

x partial derivative value = -3

x partial:h!=0;increment=h*((-1)+2*(1)*(-1)+(1)*h)

difference quotient=-3+(1)*h

x partial:quotient=-3+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(-3))=abs(1)*abs(h)<epsilon;limit=-3

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 partial derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute x partial derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute x partial 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 x partial slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute x partial slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute x partial slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute x partial slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute x partial slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute x partial 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

([4, -1, -1, 1, 0, -1], -1, -1, -3)

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.