# Whole Partial Polynomials, Fixed-x Slices and Symbolic Quotient Limits

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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-20261025-r07-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. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0001/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 2. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0002/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 3. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [0, 6, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0003/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 4. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0004/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 5. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [2, 2, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0005/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 6. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [-2, 4, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0006/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 7. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [-1, 6, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0007/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 8. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0008/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 9. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [1, 2, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0009/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 10. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [2, 4, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0010/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 11. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [-2, 6, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0011/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 12. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0012/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 13. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [0, 2, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0013/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 14. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [1, 4, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0014/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 15. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [2, 6, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0015/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 16. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [-2, 8, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0016/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 17. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [-1, 2, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0017/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 18. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0018/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 19. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, 6, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0019/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 20. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0020/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 21. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [-2, 2, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0021/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 22. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0022/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 23. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [0, 6, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0023/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 24. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y])

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0024/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 25. The Entire x-Partial Polynomial in an Explicit Basis

Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,x-partial coefficients in basis [1,x,y],[1,x,y]).

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

Differentiate symbolically to obtain [c_x,2*c_xx,c_xy], preserving zero coefficients and the basis order. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [2, 2, -1], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0025/)

Partial derivative · qf-mod25-partialdeep-partial-x-polynomial · candidate





## 26. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0026/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 27. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [-2, 0, -2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0027/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 28. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0028/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 29. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0029/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 30. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [1, 0, 4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0030/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 31. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [2, 1, -4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0031/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 32. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [3, -1, -2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0032/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 33. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0033/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 34. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0034/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 35. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [-1, -1, 4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0035/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 36. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [0, 0, -4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0036/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 37. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [1, 1, -2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0037/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 38. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [2, -1, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0038/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 39. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [3, 0, 2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0039/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 40. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [-3, 1, 4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0040/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 41. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [-2, -1, -4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0041/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 42. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [-1, 0, -2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0042/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 43. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [0, 1, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0043/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 44. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, -1, 2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0044/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 45. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0045/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 46. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [3, 1, -4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0046/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 47. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [-3, -1, -2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0047/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 48. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

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

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [-2, 0, 0], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0048/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 49. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0049/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 50. The Entire y-Partial Polynomial in the Same Coordinate Basis

Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. Give (original coefficients,y-partial coefficients in basis [1,x,y],[1,x,y]).

View answer and explanation([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y])

The coefficient list is [c_y,c_xy,2*c_yy]; do not swap x and y roles. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0050/)

Partial derivative · qf-mod25-partialdeep-partial-y-polynomial · candidate





## 51. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0051/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 52. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-11, -1, -2, 2, 0, -1], -1, [-8, -2, -1], [-2, -2], -2, 2)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], -1, [-8, -2, -1], [-2, -2], -2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0052/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 53. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], 0, [-10, -1, 0], [-1, 0], -1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0053/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 54. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 0, -1)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 0, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0054/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 55. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 5)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0055/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 56. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0056/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 57. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-6, -1, 3, 3, -1, -1], -1, [-2, 4, -1], [4, -2], 3, -2)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], -1, [-2, 4, -1], [4, -2], 3, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0057/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 58. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 0], -3, -3)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 0], -3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0058/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 59. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-4, 1, -2, 1, 1, 1], 1, [-2, -1, 1], [-1, 2], -2, -5)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], 1, [-2, -1, 1], [-1, 2], -2, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0059/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 60. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -1, -7)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -1, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0060/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 61. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 0, 0)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0061/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 62. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], -1, [4, 0, -1], [0, -2], 1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0062/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 63. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], 0, [0, 2, 0], [2, 0], 2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0063/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 64. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0064/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 65. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0065/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 66. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0066/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 67. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([4, -1, -1, 1, 0, -1], -1, [6, -1, -1], [-1, -2], -1, 1)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [6, -1, -1], [-1, -2], -1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0067/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 68. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [0, 0], 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0068/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 69. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], 1, [10, 0, 1], [0, 2], 1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0069/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 70. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0070/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 71. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-2, evaluation y=3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0071/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 72. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=-1, evaluation y=-3. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0072/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 73. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=0, evaluation y=-2. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

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

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 0], -2, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0073/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 74. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=1, evaluation y=-1. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([11, 1, -1, 4, 1, 1], 1, [16, 0, 1], [0, 2], -1, -2)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], 1, [16, 0, 1], [0, 2], -1, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0074/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 75. A Fixed-x Slice and Its y-Derivative

Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. fixed x=2, evaluation y=0. Give (coefficients,fixed x,ascending y-slice coefficients,ascending derivative coefficients,y,derivative value).

View answer and explanation([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 0, -2)

Substitute the fixed first coordinate before taking the one-variable derivative in y. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 0, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0075/)

Partial derivative · qf-mod25-partialdeep-partial-y-slice-derivative · candidate





## 76. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-12, -2, -3, 1, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0076/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 77. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0077/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 78. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0078/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 79. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-9, 1, 0, 4, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0079/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 80. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-8, 2, 1, 1, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0080/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 81. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0081/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 82. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-6, -1, 3, 3, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0082/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 83. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-5, 0, -3, 4, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0083/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 84. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-4, 1, -2, 1, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0084/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 85. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-3, 2, -1, 2, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0085/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 86. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0086/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 87. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[-1, -1, 1, 4, 1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0087/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 88. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[0, 0, 2, 1, -1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0088/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 89. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[1, 1, 3, 2, 0, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0089/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 90. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0090/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 91. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0091/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 92. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0092/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 93. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[5, 0, 0, 2, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0093/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 94. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[6, 1, 1, 3, -1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0094/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 95. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[7, 2, 2, 4, 0, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0095/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 96. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-2, y=3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0096/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 97. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=-1, y=-3; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0097/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 98. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=0, y=-2; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0098/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 99. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[11, 1, -1, 4, 1, 1] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=1, y=-1; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0099/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate





## 100. Cancel Only Nonzero h before Taking the Symbolic Slice Limit

Use the quadratic candidate coefficients=[12, 2, 0, 1, -1, 2] in fixed order [1,x,y,x²,x*y,y²]. Fix the other coordinate before taking a one-variable derivative. These symbolic polynomial identities do not replace a limit by one finite-difference sample or establish total differentiability of general functions. point x=2, y=0; keep y fixed. Give (coefficients,x,y,[constant,h coefficient] of the exact x-difference quotient,quotient=constant+h_coefficient*h for h≠0,limit,symbolic_limit_not_finite_difference).

View answer and explanation([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)

Expand u(x+h,y)−u(x,y), divide only for nonzero h and then take h→0 in the resulting polynomial. The h coefficient is c_xx and the limit is the constant quotient coefficient, not one sampled finite difference. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r07-partial-derivative-0100/)

Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate



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