Whole Partial Polynomials, Fixed-x Slices and Symbolic Quotient Limits
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Partial derivative · qf-mod25-partialdeep-slice-difference-quotient-limit · candidate
Canonical question JSONL · English JSONL · Validation report