Derivative Ontology

FLOW ALGEBRA
DERIVATIVE ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Calculus and Analysis Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/derivative-ontology-2.html
Version: 1.0 | Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)

1. ONTOLOGY IDENTITY
ontology_id: kireth:mathematical-entities:derivative
entity_type: MathematicalOperator
preferred_label: Derivative
scope: local and higher-order differentiation of real- and vector-valued functions; extensions explicitly typed

2. TYPED ENTITIES
Function: f : D -> Y
Domain: D subseteq R or open subset of R^n
Point: a in D
Increment: h in R or R^n
Derivative: f'(a), Df(a), partial_i f(a)
Tangent: linear first-order approximation
Differentiability: property of function at a point
Continuity: property of function at a point
HigherOrderDerivative: f^(n)
Jacobian: matrix representation of total derivative
Gradient: vector representing differential under Euclidean inner product

3. MATHEMATICAL SEMANTICS
For a real-valued f on an interval with a an interior point:
f'(a) = lim_{h -> 0} [f(a+h)-f(a)]/h, provided the finite two-sided limit exists.
Differentiability at a implies continuity at a; the converse is false.
At domain endpoints, one-sided derivatives must be labelled separately.
For f: U subseteq R^n -> R^m, Frechet differentiability at a means there is a linear map A such that
lim_{h -> 0} ||f(a+h)-f(a)-Ah|| / ||h|| = 0, with a+h in U.
Then Df(a)=A and its coordinate representation is the Jacobian J_f(a).
Partial derivatives alone need not imply total differentiability.

4. CORE RULES
R01 CONSTANT: D(c)=0.
R02 IDENTITY: D(x)=1.
R03 LINEARITY: D(af+bg)=aDf+bDg, for constant a,b and differentiable f,g.
R04 PRODUCT: (fg)'=f'g+fg'.
R05 QUOTIENT: (f/g)'=(f'g-fg')/g^2 where g != 0.
R06 CHAIN: (g o f)'(x)=g'(f(x))f'(x), with applicable differentiability.
R07 POWER: D(x^n)=n*x^(n-1) wherever the real-valued expression is defined and differentiable.
R08 INVERSE: (f^{-1})'(f(a))=1/f'(a), where the local inverse theorem hypotheses hold and f'(a) != 0.
R09 HIGHER_ORDER: f^(n)=D(f^(n-1)) where successive derivatives exist.
R10 MIXED_PARTIALS: partial_i partial_j f = partial_j partial_i f under sufficient regularity, e.g. continuous second partials near the point.

5. RELATIONSHIP GRAPH
Derivative --is_operator_on--> Function
Derivative --evaluated_at--> Point
Derivative --defined_by--> DifferenceQuotientLimit
Derivative --requires--> Differentiability
Differentiability --implies--> Continuity
Derivative --determines--> TangentSlope [scalar case]
TotalDerivative --represented_by--> Jacobian [finite-dimensional coordinates]
Differential --represented_by--> Gradient [scalar field with Euclidean inner product]
Derivative --generalised_by--> FrechetDerivative
Derivative --related_to--> Limit
Derivative --related_to--> Integral [fundamental theorem under stated hypotheses]

6. INFERENCE AND VALIDATION
IF f and g differentiable at a THEN f+g differentiable at a.
IF f differentiable at a THEN f continuous at a.
IF f not continuous at a THEN f not differentiable at a.
IF left and right finite derivatives exist and are equal at an interior point THEN derivative exists.
IF derivative is asserted THEN verify domain, point, limit existence and derivative type.
DO NOT infer differentiability from continuity alone.
DO NOT equate partial differentiability with total differentiability.
DO NOT apply ordinary derivative rules to distributional or weak derivatives without explicit typing.

7. SEMANTIC_EQUIVALENCE
ordinary_derivative <=> first_derivative [one-dimensional context]
f_prime(a) <=> df/dx at a [one-dimensional real-valued context]
Jacobian <=> coordinate_matrix_of_total_derivative [finite-dimensional context]
NOT_EQUIVALENT: derivative != differential_equation
NOT_EQUIVALENT: partial_derivative != total_derivative
NOT_EQUIVALENT: continuity != differentiability

8. CONFIDENCE
mathematical_axioms: 1.00 [within declared assumptions]
rule_application: conditional on hypotheses
external_page_alignment: unverified; this is a domain-grounded ontology draft

9. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; normalised line endings; exclude FLOW_HASH value itself
value: COMPUTE_ON_INGEST

10. UPDATE_MODEL
version_policy: semantic_versioning
on_change: revalidate typed entities, assumptions, rules and graph edges
provenance: preserve source URL and version history

11. AI_CONSUMPTION
parse_mode: typed_semantic_flow
required_checks: domain; codomain; derivative_kind; point; existence; regularity_assumptions
output: symbolic expression; proof obligations; inference provenance; confidence
error_policy: reject invalid derivative claims; request missing assumptions

END FLOW ALGEBRA