Original question and givens
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]).
G001 — 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]).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod25-partialdeep-partial-x-polynomial, [coefficients, domain, operation]]
After
derived = [partialdeep-partial-x-polynomial, Polynomial_partial_derivative_density]
Conditions: none
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Step identifiers and dependencies
Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001
S002 · implies
Before
source roles = [[-12, -2, -3, 1, -1, -2]]
After
derived = [1, x, y, x^2, x*y, y^2]
Conditions: none
The SIX coefficients use the full[1,x,y,x²,xy,y²]basis.Hold the OTHER coordinate fixed.This supplied polynomial supports symbolic identities,not inference of a derivative from one difference sample or total differentiability of arbitrary functions.
Step identifiers and dependencies
Rule: Partial_FULL_polynomial_basis_and_fixed_other_coordinate · Uses: G001, S001
S003 · implies
Before
source roles = [x, -12, 1]
After
derived = [-12*(0)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S002
S004 · implies
Before
source roles = [x, -2, x]
After
derived = [-2*(1)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S003
S005 · implies
Before
source roles = [x, -3, y]
After
derived = [-3*(0)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S004
S006 · implies
Before
source roles = [x, 1, x^2]
After
derived = [1*(2*x)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S005
S007 · implies
Before
source roles = [x, -1, x*y]
After
derived = [-1*(y)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S006
S008 · implies
Before
source roles = [x, -2, y^2]
After
derived = [-2*(0)]
Conditions: none
Apply the constant,power or fixed-other-coordinate product rule to EACH original monomial.Retain zero derivative terms,then collect in[1,x,y].
Step identifiers and dependencies
Rule: Partial_EACH_monomial_power_product_and_constant_rules · Uses: G001, S007
S009 · calculation
Before
Partial ORIGINAL twice square coefficient := mul(2, 1)
After
Partial ORIGINAL twice square coefficient = 2
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [x, [-12, -2, -3, 1, -1, -2], [[1, 0], [x, 1], [y, 0], [x^2, 2*x], [x*y, y], [y^2, 0]]]
After
derived = [[-2, 2, -1], [1,x,y]]
Conditions: none
Collect ALL derivative terms in the declared basis without deleting zero coefficients or swapping x/y.Re-expand to audit the derivative contribution of every original term.
Step identifiers and dependencies
Rule: Partial_COLLECT_all_coefficients_in_declared_basis · Uses: G001, S009
Result
([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.