Original question and givens
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).
G001 — 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).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod25-partialdeep-partial-y-slice-derivative, [coefficients, domain, fixed_x, operation, y]]
After
derived = [partialdeep-partial-y-slice-derivative, 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 = [[5, 0, 0, 2, 1, 0]]
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 · calculation
Before
Slice ORIGINAL c_x times fixed x := mul(0, 0)
After
Slice ORIGINAL c_x times fixed x = 0
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, S002
S004 · calculation
Before
Slice ORIGINAL fixed x squared := mul(0, 0)
After
Slice ORIGINAL fixed x squared = 0
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, S003
S005 · calculation
Before
Slice ORIGINAL c_xx times fixed x squared := mul(2, 0)
After
Slice ORIGINAL c_xx times fixed x squared = 0
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, S004
S006 · calculation
Before
Slice ORIGINAL constant terms sum := sum(5, 0, 0)
After
Slice ORIGINAL constant terms sum = 5
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, S005
S007 · calculation
Before
Slice ORIGINAL c_xy times fixed x := mul(1, 0)
After
Slice ORIGINAL c_xy times fixed x = 0
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, S006
S008 · calculation
Before
Slice ORIGINAL linear y terms sum := sum(0, 0)
After
Slice ORIGINAL linear y terms sum = 0
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, S007
S009 · calculation
Before
Slice ORIGINAL twice c_yy := mul(2, 0)
After
Slice ORIGINAL twice c_yy = 0
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 = [[5, 0, 0, 2, 1, 0], 0]
After
derived = [[5, 0, 0], [0, 0]]
Conditions: none
FIRST fix x and collect the WHOLE y-polynomial s0+s1*y+s2*y².THEN differentiate to s1+2*s2*y.Do not substitute BOTH coordinates into a constant before differentiating.
Step identifiers and dependencies
Rule: Partial_FIXED_x_whole_y_polynomial_THEN_derivative · Uses: G001, S009
S011 · calculation
Before
Slice ORIGINAL derivative linear term at y := mul(0, 0)
After
Slice ORIGINAL derivative linear term at y = 0
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, S010
S012 · calculation
Before
Slice ORIGINAL full derivative value := sum(0, 0)
After
Slice ORIGINAL full derivative value = 0
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, S011
Result
([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [0, 0], 0, 0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.