# Question 11530: Differentiate in the First Coordinate while Holding the Second Fixed [Return to catalogue page 58](https://ocme.evemisslab.com/en/questions/page/58/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r04-partial-derivative-foundations/#q-20261013-advance-20261024-r04-partial-derivative-0005) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0022-11530 · q-20261013-advance-20261024-r04-partial-derivative-0005 ## 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](https://ocme.evemisslab.com/en/mko/mko-partial-derivative/) ## Step-by-step derivation S001 · impliesBefore original coefficients=[-8, 2, 1, 1, 0, -1];basis=[1,x,y,x^2,xy,y^2] After u(x,y)=(-8)+(2)*x+(1)*y+(1)*x^2+(0)*x*y+(-1)*y^2 Conditions: none Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic. Step identifiers and dependenciesRule: bivariate_scalar_polynomial_basis_roles · Uses: G001 S002 · calculationBefore x partial slice fixed coordinate squared := mul(1, 1) After x partial slice fixed coordinate squared = 1 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore x partial slice fixed linear term := mul(1, 1) After x partial slice fixed linear term = 1 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore x partial slice fixed square term := mul(-1, 1) After x partial slice fixed square term = -1 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore x partial slice constant coefficient := sum(-8, 1, -1) After x partial slice constant coefficient = -8 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore x partial slice cross contribution := mul(0, 1) After x partial slice cross contribution = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore x partial slice linear coefficient := sum(2, 0) After x partial slice linear coefficient = 2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore x partial slice:vary x;fix y=1;original coefficients=[-8, 2, 1, 1, 0, -1] After ascending x slice coefficients=[-8, 2, 1] Conditions: none Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together. Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S007 S009 · impliesBefore x partial:g(t)=(-8)+(2)*t+(1)*t^2;point=2;h!=0 After g(2+h)-g(2)=(2)*h+(1)*(2*(2)*h+h^2) Conditions: none Cancel constants and expand(z+h)^2−z^2=2zh+h^2.Do not substitute h=0 at this stage. Step identifiers and dependenciesRule: slice_increment_exact_expansion · Uses: G001, S008 S010 · calculationBefore x partial twice quadratic coefficient := mul(2, 1) After x partial twice quadratic coefficient = 2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore x partial derivative point term := mul(2, 2) After x partial derivative point term = 4 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore x partial derivative value := sum(2, 4) After x partial derivative value = 6 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore x partial:h!=0;increment=h*((2)+2*(1)*(2)+(1)*h) After difference quotient=6+(1)*h Conditions: none Factor out the whole h and cancel only under h≠0.The quotient still has D*h;a finite difference is not automatically the derivative. Step identifiers and dependenciesRule: cancel_only_nonzero_increment · Uses: G001, S012 S014 · impliesBefore x partial:quotient=6+(1)*h;epsilon>0;delta=epsilon/1 After 0