Step-by-step derivation
S001 · implies
Before
source operation=higher-order-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate]
Conditions: none
Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL low-to-high coefficients=[11, 22, 33, 44, 55, 66];declared order=3
After
declared degree BOUND=5=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list
Conditions: none
Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions.
Step identifiers and dependencies
Rule: higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · Uses: G001, S001
S003 · calculation
Before
Higher derivative level1 ORIGINAL power1 coefficient product := mul(22, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 22
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Higher derivative level1 ORIGINAL power2 coefficient product := mul(33, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 66
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Higher derivative level1 ORIGINAL power3 coefficient product := mul(44, 3)
After
Higher derivative level1 ORIGINAL power3 coefficient product = 132
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Higher derivative level1 ORIGINAL power4 coefficient product := mul(55, 4)
After
Higher derivative level1 ORIGINAL power4 coefficient product = 220
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Higher derivative level1 ORIGINAL power5 coefficient product := mul(66, 5)
After
Higher derivative level1 ORIGINAL power5 coefficient product = 330
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
previous derivative order=0;actual low-to-high list=[11, 22, 33, 44, 55, 66];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 22, 1, 22], [2, 33, 2, 66], [3, 44, 3, 132], [4, 55, 4, 220], [5, 66, 5, 330]];next raw list=[22, 66, 132, 220, 330];all intermediate products SAFE
Conditions: none
At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.
Step identifiers and dependencies
Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S007
S009 · calculation
Before
Higher derivative level2 ORIGINAL power1 coefficient product := mul(66, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 66
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Higher derivative level2 ORIGINAL power2 coefficient product := mul(132, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 264
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Higher derivative level2 ORIGINAL power3 coefficient product := mul(220, 3)
After
Higher derivative level2 ORIGINAL power3 coefficient product = 660
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
Higher derivative level2 ORIGINAL power4 coefficient product := mul(330, 4)
After
Higher derivative level2 ORIGINAL power4 coefficient product = 1320
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · implies
Before
previous derivative order=1;actual low-to-high list=[22, 66, 132, 220, 330];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 66, 1, 66], [2, 132, 2, 264], [3, 220, 3, 660], [4, 330, 4, 1320]];next raw list=[66, 264, 660, 1320];all intermediate products SAFE
Conditions: none
At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.
Step identifiers and dependencies
Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S012
S014 · calculation
Before
Higher derivative level3 ORIGINAL power1 coefficient product := mul(264, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 264
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Higher derivative level3 ORIGINAL power2 coefficient product := mul(660, 2)
After
Higher derivative level3 ORIGINAL power2 coefficient product = 1320
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Higher derivative level3 ORIGINAL power3 coefficient product := mul(1320, 3)
After
Higher derivative level3 ORIGINAL power3 coefficient product = 3960
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · implies
Before
previous derivative order=2;actual low-to-high list=[66, 264, 660, 1320];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 264, 1, 264], [2, 660, 2, 1320], [3, 1320, 3, 3960]];next raw list=[264, 1320, 3960];all intermediate products SAFE
Conditions: none
At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.
Step identifiers and dependencies
Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S016
S018 · implies
Before
ORIGINAL coefficients=[11, 22, 33, 44, 55, 66];order=3;declared degree bound=5
After
final derivative coefficients=[264, 1320, 3960];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish
Conditions: none
Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree.
Step identifiers and dependencies
Rule: higher_falling_factorial_rule_all_orders_polynomial_only · Uses: G001, S017