Derivation

Derivation is the act or process of obtaining something from a source through a series of steps or transformations. In mathematics, the term often refers to the derivative concept or the process of determining a derivative. The derivative of a function describes its instantaneous rate of change; formally, f'(x) = lim h->0 [f(x+h) - f(x)]/h. Derivation is also used for the process of deducing a formula or theorem from prior principles or rules, such as deriving integration formulas or laws of physics from fundamental assumptions.

In linguistics, derivation refers to forming a new word or grammatical category from an existing word, often by affixation, compounding, or back-formation. This contrasts with inflection, which marks grammatical features without changing the word class, e.g., "talk" -> "talked" (inflection) versus "happiness" from "happy" (derivation). Examples include "unbelievable," "communication," and "rewrite."

In logic and computer science, a derivation is a sequence of inference steps that leads from axioms or assumptions to a conclusion or theorem, or, in formal grammars, a step-by-step replacement of nonterminals to derive a string of terminal symbols. In these contexts, derivations illustrate how complex results emerge from basic rules.

Derivation is used more generally to denote any method of reasoning or transformation that yields new knowledge from existing information. Across disciplines, it reflects the transfer of structure, properties, or meanings from a source to a derived form.