Step-by-step derivation
S001 · implies
Before
source operation=higher-zero-beyond-degree;original snapshot and mathematics retained
After
explicit ordered source output fields=[declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, 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=[14, 28, 42, 56, 70];declared order=6
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(28, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 28
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(42, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 84
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(56, 3)
After
Higher derivative level1 ORIGINAL power3 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, S004
S006 · calculation
Before
Higher derivative level1 ORIGINAL power4 coefficient product := mul(70, 4)
After
Higher derivative level1 ORIGINAL power4 coefficient product = 280
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=[14, 28, 42, 56, 70];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 28, 1, 28], [2, 42, 2, 84], [3, 56, 3, 168], [4, 70, 4, 280]];next raw list=[28, 84, 168, 280];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(84, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 84
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(168, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 336
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(280, 3)
After
Higher derivative level2 ORIGINAL power3 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, S009
S011 · implies
Before
previous derivative order=1;actual low-to-high list=[28, 84, 168, 280];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 84, 1, 84], [2, 168, 2, 336], [3, 280, 3, 840]];next raw list=[84, 336, 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, S010
S012 · calculation
Before
Higher derivative level3 ORIGINAL power1 coefficient product := mul(336, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 336
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(840, 2)
After
Higher derivative level3 ORIGINAL power2 coefficient product = 1680
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=[84, 336, 840];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 336, 1, 336], [2, 840, 2, 1680]];next raw list=[336, 1680];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(1680, 1)
After
Higher derivative level4 ORIGINAL power1 coefficient product = 1680
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=[336, 1680];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1680, 1, 1680]];next raw list=[1680];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=[1680];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
previous derivative order=5;actual low-to-high list=[];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, S017
S019 · implies
Before
ORIGINAL coefficients=[14, 28, 42, 56, 70];order=6;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, S018
S020 · calculation
Before
Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, 1)
After
Higher derivative Horner power0 previous value times ORIGINAL point = 0
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, S019
S021 · calculation
Before
Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)
After
Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0
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, S020