# Fixed-Coordinate Polynomial Partials and Ordered Gradients

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261024-r04-partial-derivative-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-12, -2, -3, 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. Evaluate at x=-2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-12, -2, -3, 1, -1, -1], -2, -3, -3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0001/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 2. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-11, -1, -2, 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=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-11, -1, -2, 1, 0, -1], -1, -2, -3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0002/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 3. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-10, 0, -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. Evaluate at x=0, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-10, 0, -1, 1, 1, -1], 0, -1, -1)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0003/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 4. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-9, 1, 0, 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. Evaluate at x=1, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-9, 1, 0, 1, -1, -1], 1, 0, 3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0004/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 5. Differentiate in the First Coordinate while Holding the Second Fixed

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).

View answer and explanation([-8, 2, 1, 1, 0, -1], 2, 1, 6)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0005/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 6. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-7, -2, 2, 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. Evaluate at x=-2, y=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-7, -2, 2, 1, 1, -1], -2, 2, -4)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0006/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 7. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-6, -1, 3, 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. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-6, -1, 3, 1, -1, -1], -1, 3, -6)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0007/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 8. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-5, 0, -3, 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=0, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-5, 0, -3, 1, 0, -1], 0, -3, 0)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0008/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 9. Differentiate in the First Coordinate while Holding the Second Fixed

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

View answer and explanation([-4, 1, -2, 1, 1, -1], 1, -2, 1)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0009/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 10. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[-3, 2, -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. Evaluate at x=2, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([-3, 2, -1, 1, -1, -1], 2, -1, 7)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0010/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 11. Differentiate in the First Coordinate while Holding the Second Fixed

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

View answer and explanation([-2, -2, 0, 1, 0, -1], -2, 0, -6)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0011/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 12. Differentiate in the First Coordinate while Holding the Second Fixed

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

View answer and explanation([-1, -1, 1, 1, 1, -1], -1, 1, -2)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0012/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 13. Differentiate in the First Coordinate while Holding the Second Fixed

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

View answer and explanation([0, 0, 2, 1, -1, -1], 0, 2, -2)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0013/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 14. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[1, 1, 3, 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=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([1, 1, 3, 1, 0, -1], 1, 3, 3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0014/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 15. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[2, 2, -3, 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. Evaluate at x=2, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([2, 2, -3, 1, 1, -1], 2, -3, 3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0015/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 16. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[3, -2, -2, 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. Evaluate at x=-2, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([3, -2, -2, 1, -1, -1], -2, -2, -4)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0016/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 17. Differentiate in the First Coordinate while Holding the Second Fixed

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).

View answer and explanation([4, -1, -1, 1, 0, -1], -1, -1, -3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0017/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 18. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[5, 0, 0, 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. Evaluate at x=0, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([5, 0, 0, 1, 1, -1], 0, 0, 0)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0018/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 19. Differentiate in the First Coordinate while Holding the Second Fixed

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. Evaluate at x=1, y=1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([6, 1, 1, 1, -1, -1], 1, 1, 2)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0019/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 20. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[7, 2, 2, 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=2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([7, 2, 2, 1, 0, -1], 2, 2, 6)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0020/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 21. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[8, -2, 3, 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. Evaluate at x=-2, y=3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([8, -2, 3, 1, 1, -1], -2, 3, -3)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0021/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 22. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[9, -1, -3, 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. Evaluate at x=-1, y=-3. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([9, -1, -3, 1, -1, -1], -1, -3, 0)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0022/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 23. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[10, 0, -2, 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=0, y=-2. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([10, 0, -2, 1, 0, -1], 0, -2, 0)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0023/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 24. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[11, 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. Evaluate at x=1, y=-1. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([11, 1, -1, 1, 1, -1], 1, -1, 2)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0024/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 25. Differentiate in the First Coordinate while Holding the Second Fixed

Use coefficients=[12, 2, 0, 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. Evaluate at x=2, y=0. Give (coefficients,x,y,c_x+2*c_xx*x+c_xy*y).

View answer and explanation([12, 2, 0, 1, -1, -1], 2, 0, 6)

Differentiate only the x-dependent monomials while treating y as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0025/)

Partial derivative · qf-mod24-partial-partial-x · candidate





## 26. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-12, -2, -3, 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. Evaluate at x=-2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-12, -2, -3, 1, -1, -1], -2, -3, 5)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0026/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 27. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-11, -1, -2, 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=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-11, -1, -2, 1, 0, -1], -1, -2, 2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0027/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 28. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-10, 0, -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. Evaluate at x=0, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-10, 0, -1, 1, 1, -1], 0, -1, 1)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0028/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 29. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-9, 1, 0, 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. Evaluate at x=1, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-9, 1, 0, 1, -1, -1], 1, 0, -1)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0029/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 30. Differentiate in the Second Coordinate while Holding the First Fixed

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_y+c_xy*x+2*c_yy*y).

View answer and explanation([-8, 2, 1, 1, 0, -1], 2, 1, -1)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0030/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 31. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-7, -2, 2, 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. Evaluate at x=-2, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-7, -2, 2, 1, 1, -1], -2, 2, -4)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0031/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 32. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-6, -1, 3, 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. Evaluate at x=-1, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-6, -1, 3, 1, -1, -1], -1, 3, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0032/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 33. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-5, 0, -3, 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=0, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-5, 0, -3, 1, 0, -1], 0, -3, 3)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0033/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 34. Differentiate in the Second Coordinate while Holding the First Fixed

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

View answer and explanation([-4, 1, -2, 1, 1, -1], 1, -2, 3)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0034/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 35. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-3, 2, -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. Evaluate at x=2, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-3, 2, -1, 1, -1, -1], 2, -1, -1)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0035/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 36. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-2, -2, 0, 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=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-2, -2, 0, 1, 0, -1], -2, 0, 0)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0036/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 37. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[-1, -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. Evaluate at x=-1, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([-1, -1, 1, 1, 1, -1], -1, 1, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0037/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 38. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[0, 0, 2, 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. Evaluate at x=0, y=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([0, 0, 2, 1, -1, -1], 0, 2, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0038/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 39. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[1, 1, 3, 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=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([1, 1, 3, 1, 0, -1], 1, 3, -3)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0039/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 40. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[2, 2, -3, 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. Evaluate at x=2, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([2, 2, -3, 1, 1, -1], 2, -3, 5)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0040/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 41. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[3, -2, -2, 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. Evaluate at x=-2, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([3, -2, -2, 1, -1, -1], -2, -2, 4)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0041/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 42. Differentiate in the Second Coordinate while Holding the First Fixed

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).

View answer and explanation([4, -1, -1, 1, 0, -1], -1, -1, 1)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0042/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 43. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[5, 0, 0, 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. Evaluate at x=0, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([5, 0, 0, 1, 1, -1], 0, 0, 0)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0043/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 44. Differentiate in the Second Coordinate while Holding the First Fixed

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. Evaluate at x=1, y=1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([6, 1, 1, 1, -1, -1], 1, 1, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0044/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 45. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[7, 2, 2, 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=2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([7, 2, 2, 1, 0, -1], 2, 2, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0045/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 46. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[8, -2, 3, 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. Evaluate at x=-2, y=3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([8, -2, 3, 1, 1, -1], -2, 3, -5)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0046/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 47. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[9, -1, -3, 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. Evaluate at x=-1, y=-3. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([9, -1, -3, 1, -1, -1], -1, -3, 4)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0047/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 48. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[10, 0, -2, 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=0, y=-2. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([10, 0, -2, 1, 0, -1], 0, -2, 2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0048/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 49. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[11, 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. Evaluate at x=1, y=-1. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([11, 1, -1, 1, 1, -1], 1, -1, 2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0049/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 50. Differentiate in the Second Coordinate while Holding the First Fixed

Use coefficients=[12, 2, 0, 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. Evaluate at x=2, y=0. Give (coefficients,x,y,c_y+c_xy*x+2*c_yy*y).

View answer and explanation([12, 2, 0, 1, -1, -1], 2, 0, -2)

Differentiate only the y-dependent monomials while treating x as constant. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0050/)

Partial derivative · qf-mod24-partial-partial-y · candidate





## 51. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-12, -2, -3, 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. x=-2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0051/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 52. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-11, -1, -2, 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. x=-1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0052/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 53. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-10, 0, -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. x=0, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0053/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 54. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-9, 1, 0, 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. x=1, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0054/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 55. Collect the Two Coordinate Partials in Their Declared Order

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. x=2, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0055/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 56. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-7, -2, 2, 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. x=-2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0056/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 57. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-6, -1, 3, 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. x=-1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0057/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 58. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-5, 0, -3, 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. x=0, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0058/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 59. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-4, 1, -2, 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. x=1, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0059/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 60. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-3, 2, -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. x=2, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0060/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 61. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-2, -2, 0, 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. x=-2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0061/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 62. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[-1, -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. x=-1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0062/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 63. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[0, 0, 2, 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. x=0, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0063/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 64. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[1, 1, 3, 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. x=1, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, [3, -3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0064/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 65. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[2, 2, -3, 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. x=2, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, [3, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0065/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 66. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[3, -2, -2, 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. x=-2, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0066/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 67. Collect the Two Coordinate Partials in Their Declared Order

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. x=-1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0067/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 68. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[5, 0, 0, 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. x=0, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0068/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 69. Collect the Two Coordinate Partials in Their Declared Order

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. x=1, y=1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, [2, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0069/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 70. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[7, 2, 2, 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. x=2, y=2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, [6, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0070/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 71. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[8, -2, 3, 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. x=-2, y=3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0071/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 72. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[9, -1, -3, 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. x=-1, y=-3. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, [0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0072/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 73. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[10, 0, -2, 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. x=0, y=-2. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0073/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 74. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[11, 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. x=1, y=-1. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, [2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0074/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 75. Collect the Two Coordinate Partials in Their Declared Order

Use coefficients=[12, 2, 0, 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. x=2, y=0. Give (coefficients,x,y,[partial_x,partial_y]) in this fixed real coordinate model.

View answer and explanation([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])

Compute the two partials separately and collect them in x-then-y order. This ordered coordinate calculation does not establish a general differentiability theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, [6, -2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0075/)

Partial derivative · qf-mod24-partial-gradient · candidate





## 76. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-12, -2, -3, 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=-2, evaluation x=-3. 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.

View answer and explanation([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0076/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 77. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-11, -1, -2, 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. fixed y=-1, evaluation x=-2. 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.

View answer and explanation([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0077/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 78. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-10, 0, -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=0, 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.

View answer and explanation([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0078/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 79. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-9, 1, 0, 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=0. 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.

View answer and explanation([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0079/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 80. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

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. fixed y=2, 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.

View answer and explanation([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0080/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 81. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-7, -2, 2, 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=-2, evaluation x=2. 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.

View answer and explanation([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0081/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 82. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-6, -1, 3, 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=3. 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.

View answer and explanation([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0082/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 83. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-5, 0, -3, 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. fixed y=0, evaluation x=-3. 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.

View answer and explanation([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0083/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 84. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-4, 1, -2, 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=-2. 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.

View answer and explanation([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0084/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 85. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-3, 2, -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=2, 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.

View answer and explanation([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0085/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 86. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-2, -2, 0, 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. fixed y=-2, evaluation x=0. 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.

View answer and explanation([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0086/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 87. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[-1, -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.

View answer and explanation([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0087/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 88. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[0, 0, 2, 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=0, evaluation x=2. 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.

View answer and explanation([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0088/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 89. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[1, 1, 3, 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. fixed y=1, evaluation x=3. 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.

View answer and explanation([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0089/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 90. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[2, 2, -3, 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=2, evaluation x=-3. 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.

View answer and explanation([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0090/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 91. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[3, -2, -2, 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=-2, evaluation x=-2. 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.

View answer and explanation([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0091/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 92. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

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. 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.

View answer and explanation([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0092/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 93. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[5, 0, 0, 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=0, evaluation x=0. 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.

View answer and explanation([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0093/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 94. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

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.

View answer and explanation([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0094/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 95. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[7, 2, 2, 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. fixed y=2, evaluation x=2. 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.

View answer and explanation([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0095/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 96. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[8, -2, 3, 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=-2, evaluation x=3. 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.

View answer and explanation([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0096/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 97. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[9, -1, -3, 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=-3. 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.

View answer and explanation([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0097/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 98. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[10, 0, -2, 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. fixed y=0, evaluation x=-2. 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.

View answer and explanation([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0098/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 99. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[11, 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.

View answer and explanation([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0099/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate





## 100. A Fixed-y Polynomial Slice and Its Exact One-Variable Derivative

Use coefficients=[12, 2, 0, 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=2, evaluation x=0. 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.

View answer and explanation([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)

Substitute the fixed second coordinate before differentiating the resulting x-polynomial; keep coefficient order and zero coefficients. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r04-partial-derivative-0100/)

Partial derivative · qf-mod24-partial-x-slice-derivative · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r04-partial-derivative-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261024-r04-partial-derivative-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r04-partial-derivative-foundations/validation-report.json)