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=[25, 50, 75, 100];declared order=4
After
declared degree BOUND=3=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(50, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 50
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(75, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 150
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(100, 3)
After
Higher derivative level1 ORIGINAL power3 coefficient product = 300
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 · implies
Before
previous derivative order=0;actual low-to-high list=[25, 50, 75, 100];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 50, 1, 50], [2, 75, 2, 150], [3, 100, 3, 300]];next raw list=[50, 150, 300];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, S005
S007 · calculation
Before
Higher derivative level2 ORIGINAL power1 coefficient product := mul(150, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 150
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 level2 ORIGINAL power2 coefficient product := mul(300, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 600
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=1;actual low-to-high list=[50, 150, 300];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 150, 1, 150], [2, 300, 2, 600]];next raw list=[150, 600];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 level3 ORIGINAL power1 coefficient product := mul(600, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 600
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=2;actual low-to-high list=[150, 600];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 600, 1, 600]];next raw list=[600];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 · implies
Before
previous derivative order=3;actual low-to-high list=[600];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, S011
S013 · implies
Before
ORIGINAL coefficients=[25, 50, 75, 100];order=4;declared degree bound=3
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, S012
S014 · calculation
Before
Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, 2)
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, S013
S015 · 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, S014