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=[4, 8, 12, 16, 20, 24, 28];declared order=3
After
declared degree BOUND=6=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(8, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 8
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(12, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 24
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(16, 3)
After
Higher derivative level1 ORIGINAL power3 coefficient product = 48
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(20, 4)
After
Higher derivative level1 ORIGINAL power4 coefficient product = 80
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(24, 5)
After
Higher derivative level1 ORIGINAL power5 coefficient product = 120
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 · calculation
Before
Higher derivative level1 ORIGINAL power6 coefficient product := mul(28, 6)
After
Higher derivative level1 ORIGINAL power6 coefficient product = 168
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, S007
S009 · implies
Before
previous derivative order=0;actual low-to-high list=[4, 8, 12, 16, 20, 24, 28];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 8, 1, 8], [2, 12, 2, 24], [3, 16, 3, 48], [4, 20, 4, 80], [5, 24, 5, 120], [6, 28, 6, 168]];next raw list=[8, 24, 48, 80, 120, 168];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, S008
S010 · calculation
Before
Higher derivative level2 ORIGINAL power1 coefficient product := mul(24, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 24
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 power2 coefficient product := mul(48, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 96
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 power3 coefficient product := mul(80, 3)
After
Higher derivative level2 ORIGINAL power3 coefficient product = 240
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 · calculation
Before
Higher derivative level2 ORIGINAL power4 coefficient product := mul(120, 4)
After
Higher derivative level2 ORIGINAL power4 coefficient product = 480
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, S012
S014 · calculation
Before
Higher derivative level2 ORIGINAL power5 coefficient product := mul(168, 5)
After
Higher derivative level2 ORIGINAL power5 coefficient product = 840
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 · implies
Before
previous derivative order=1;actual low-to-high list=[8, 24, 48, 80, 120, 168];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 24, 1, 24], [2, 48, 2, 96], [3, 80, 3, 240], [4, 120, 4, 480], [5, 168, 5, 840]];next raw list=[24, 96, 240, 480, 840];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, S014
S016 · calculation
Before
Higher derivative level3 ORIGINAL power1 coefficient product := mul(96, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 96
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 · calculation
Before
Higher derivative level3 ORIGINAL power2 coefficient product := mul(240, 2)
After
Higher derivative level3 ORIGINAL power2 coefficient product = 480
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, S016
S018 · calculation
Before
Higher derivative level3 ORIGINAL power3 coefficient product := mul(480, 3)
After
Higher derivative level3 ORIGINAL power3 coefficient product = 1440
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, S017
S019 · calculation
Before
Higher derivative level3 ORIGINAL power4 coefficient product := mul(840, 4)
After
Higher derivative level3 ORIGINAL power4 coefficient product = 3360
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, S018
S020 · implies
Before
previous derivative order=2;actual low-to-high list=[24, 96, 240, 480, 840];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 96, 1, 96], [2, 240, 2, 480], [3, 480, 3, 1440], [4, 840, 4, 3360]];next raw list=[96, 480, 1440, 3360];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, S019
S021 · implies
Before
ORIGINAL coefficients=[4, 8, 12, 16, 20, 24, 28];order=3;declared degree bound=6
After
final derivative coefficients=[96, 480, 1440, 3360];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, S020