Question 11567: Differentiate in the Second Coordinate while Holding the First 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_y+c_xy*x+2*c_yy*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_y+c_xy*x+2*c_yy*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

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

y partial slice fixed coordinate squared = 1

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

y partial slice fixed linear term = 1

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

y partial slice fixed square term = 1

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

y partial slice constant coefficient = 6

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

y partial slice cross contribution = 0

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

y partial slice linear coefficient = -1

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

ascending y slice coefficients=[6, -1, -1]

y partial:g(t)=(6)+(-1)*t+(-1)*t^2;point=-1;h!=0

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

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

y partial twice quadratic coefficient = -2

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

y partial derivative point term = 2

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

y partial derivative value = 1

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

difference quotient=1+(-1)*h

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

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

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 y partial derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute y partial derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute y 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 y partial slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute y partial slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute y partial slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute y partial slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute y partial slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute y 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, 1)

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.