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=[17, 34, 51, 68];declared order=5
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(34, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 34
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(51, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 102
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(68, 3)
After
Higher derivative level1 ORIGINAL power3 coefficient product = 204
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=[17, 34, 51, 68];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 34, 1, 34], [2, 51, 2, 102], [3, 68, 3, 204]];next raw list=[34, 102, 204];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(102, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 102
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(204, 2)
After
Higher derivative level2 ORIGINAL power2 coefficient product = 408
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=[34, 102, 204];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 102, 1, 102], [2, 204, 2, 408]];next raw list=[102, 408];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(408, 1)
After
Higher derivative level3 ORIGINAL power1 coefficient product = 408
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=[102, 408];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 408, 1, 408]];next raw list=[408];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=[408];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
previous derivative order=4;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, S012
S014 · implies
Before
ORIGINAL coefficients=[17, 34, 51, 68];order=5;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, S013
S015 · 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, S014
S016 · 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, S015