Question 11530: Differentiate in the First Coordinate while Holding the Second Fixed

Original question and givens

Use coefficients=[-8, 2, 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=2, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

• G001 — Use coefficients=[-8, 2, 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=2, 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=[-8, 2, 1, 1, 0, -1];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-8)+(2)*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(-8, 1, -1)

x partial slice constant coefficient = -8

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

x partial slice cross contribution = 0

x partial slice linear coefficient := sum(2, 0)

x partial slice linear coefficient = 2

x partial slice:vary x;fix y=1;original coefficients=[-8, 2, 1, 1, 0, -1]

ascending x slice coefficients=[-8, 2, 1]

x partial:g(t)=(-8)+(2)*t+(1)*t^2;point=2;h!=0

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

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

x partial twice quadratic coefficient = 2

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

x partial derivative point term = 4

x partial derivative value := sum(2, 4)

x partial derivative value = 6

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

difference quotient=6+(1)*h

x partial:quotient=6+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(6))=abs(1)*abs(h)<epsilon;limit=6

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

([-8, 2, 1, 1, 0, -1], 2, 1, 6)

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.