Step-by-step derivation
S001 · implies
Before
source roles = [qf-derivative-rule-scalar, derivative_scalar, domain, reals]
After
derived = [[integer_derivative_value]]
Conditions: none
Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.
Step identifiers and dependencies
Rule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001
S002 · implies
Before
source roles = [f OUTER or first, [10, 23, 0, 3]]
After
derived = [[10, 23, 0, 3]]
Conditions: none
Check RAW ascending coefficient arity1..4 and EVERY SAFE integer BEFORE trimming trailing zeros,retaining[0].The GIVEN whole polynomial law holds on R;sample values alone do not establish differentiability.
Step identifiers and dependencies
Rule: Derivative_RAW_ascending_arity_BEFORE_trim_whole_polynomial_R · Uses: G001, S001
S003 · calculation
Before
f derivative coefficient power1 := mul(1, 23)
After
f derivative coefficient power1 = 23
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
f derivative coefficient power2 := mul(2, 0)
After
f derivative coefficient power2 = 0
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
f derivative coefficient power3 := mul(3, 3)
After
f derivative coefficient power3 = 9
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [f, [10, 23, 0, 3]]
After
derived = [[23, 0, 9]]
Conditions: none
For EVERY term use(a_k x^k)′=k a_k x^(k−1);a constant differentiates to0.Keep ALL higher terms without reapplying raw-vector bounds.
Step identifiers and dependencies
Rule: Derivative_POWER_rule_EVERY_nonconstant_term_constant_derivative_zero · Uses: G001, S005
S007 · calculation
Before
f derivative at ORIGINAL x Horner power2 times argument := mul(0, 0)
After
f derivative at ORIGINAL x Horner power2 times argument = 0
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
f derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 9)
After
f derivative at ORIGINAL x Horner power2 plus coefficient = 9
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
f derivative at ORIGINAL x Horner power1 times argument := mul(9, 0)
After
f derivative at ORIGINAL x Horner power1 times argument = 0
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
f derivative at ORIGINAL x Horner power1 plus coefficient := sum(0, 0)
After
f derivative at ORIGINAL x Horner power1 plus coefficient = 0
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
f derivative at ORIGINAL x Horner power0 times argument := mul(0, 0)
After
f derivative at ORIGINAL x Horner power0 times argument = 0
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
f derivative at ORIGINAL x Horner power0 plus coefficient := sum(0, 23)
After
f derivative at ORIGINAL x Horner power0 plus coefficient = 23
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · implies
Before
source roles = [f derivative at ORIGINAL x, [23, 0, 9], 0]
After
derived = [23]
Conditions: none
Evaluate ascending coefficients by Horner from the highest power,multiplying the DECLARED argument then adding the ORIGINAL coefficient at every step,including zeros.Do not replace an inner value with original x.
Step identifiers and dependencies
Rule: Derivative_HORNER_ALL_coefficients_AT_declared_argument · Uses: G001, S012
S014 · calculation
Before
Derivative scalar ORIGINAL c times fprime := mul(3, 23)
After
Derivative scalar ORIGINAL c times fprime = 69
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · implies
Before
source roles = [[10, 23, 0, 3], 3, 0, [23, 0, 9]]
After
derived = [23, 69]
Conditions: none
The scalar rule is(c f)′=c f′,including zero/negative c.EVEN at c=0 validate the complete RAW f and x first;do not short-circuit invalid fields.
Step identifiers and dependencies
Rule: Derivative_SCALAR_rule_ALL_raw_validated_even_c0 · Uses: G001, S014