Step-by-step derivation
S001 · implies
Before
source operation=higher-polynomial-nth-derivative;original snapshot and mathematics retained
After
explicit ordered source output fields=[coefficients, order, derivative_coefficients, point, value, 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=[9, 18, 27, 36];declared order=3
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(18, 1)
After
Higher derivative level1 ORIGINAL power1 coefficient product = 18
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(27, 2)
After
Higher derivative level1 ORIGINAL power2 coefficient product = 54
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(36, 3)
After
Higher derivative level1 ORIGINAL power3 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, S004
S006 · implies
Before
previous derivative order=0;actual low-to-high list=[9, 18, 27, 36];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 18, 1, 18], [2, 27, 2, 54], [3, 36, 3, 108]];next raw list=[18, 54, 108];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(54, 1)
After
Higher derivative level2 ORIGINAL power1 coefficient product = 54
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(108, 2)
After
Higher derivative level2 ORIGINAL power2 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, S007
S009 · implies
Before
previous derivative order=1;actual low-to-high list=[18, 54, 108];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 54, 1, 54], [2, 108, 2, 216]];next raw list=[54, 216];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(216, 1)
After
Higher derivative level3 ORIGINAL power1 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, S009
S011 · implies
Before
previous derivative order=2;actual low-to-high list=[54, 216];drop ONLY its constant term
After
ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 216, 1, 216]];next raw list=[216];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
ORIGINAL coefficients=[9, 18, 27, 36];order=3;declared degree bound=3
After
final derivative coefficients=[216];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, S011
S013 · 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, S012
S014 · calculation
Before
Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 216)
After
Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 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, S013