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=[18, 36, 54, 72, 90];declared order=5
After
declared degree BOUND=4=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(36, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 36
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(54, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 108
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(72, 3)
After
Higher derivative level1 ORIGINAL power3 coefficient product = 216
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(90, 4)
After
Higher derivative level1 ORIGINAL power4 coefficient product = 360
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 · implies
Before
previous derivative order=0;actual low-to-high list=[18, 36, 54, 72, 90];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 36, 1, 36], [2, 54, 2, 108], [3, 72, 3, 216], [4, 90, 4, 360]];next raw list=[36, 108, 216, 360];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, S006
S008 · calculation
Before
Higher derivative level2 ORIGINAL power1 coefficient product := mul(108, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 108
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 · calculation
Before
Higher derivative level2 ORIGINAL power2 coefficient product := mul(216, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 432
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 power3 coefficient product := mul(360, 3)
After
Higher derivative level2 ORIGINAL power3 coefficient product = 1080
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 · implies
Before
previous derivative order=1;actual low-to-high list=[36, 108, 216, 360];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 108, 1, 108], [2, 216, 2, 432], [3, 360, 3, 1080]];next raw list=[108, 432, 1080];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, S010
S012 · calculation
Before
Higher derivative level3 ORIGINAL power1 coefficient product := mul(432, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 432
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 level3 ORIGINAL power2 coefficient product := mul(1080, 2)
After
Higher derivative level3 ORIGINAL power2 coefficient product = 2160
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 · implies
Before
previous derivative order=2;actual low-to-high list=[108, 432, 1080];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 432, 1, 432], [2, 1080, 2, 2160]];next raw list=[432, 2160];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, S013
S015 · calculation
Before
Higher derivative level4 ORIGINAL power1 coefficient product := mul(2160, 1)
After
Higher derivative level4 ORIGINAL power1 coefficient product = 2160
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 · implies
Before
previous derivative order=3;actual low-to-high list=[432, 2160];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 2160, 1, 2160]];next raw list=[2160];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, S015
S017 · implies
Before
previous derivative order=4;actual low-to-high list=[2160];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];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=[18, 36, 54, 72, 90];order=5;declared degree bound=4
After
final derivative coefficients=[0];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