derivate

Derivate is a term used in several disciplines to describe something that is derived from another thing. In English, the most common form is derivative, and derivate may appear as a variant spelling or in other languages’ usage. In many contexts, the term indicates a relationship of dependence on underlying sources.

In calculus, the derivative of a function f with respect to a variable x measures the instantaneous rate of change of f as x changes. It is defined as the limit of the average rate of change as the increment approaches zero: f'(x) = lim_{h→0} (f(x+h) − f(x))/h. The derivative represents the slope of the function’s graph and underpins many rules of differentiation, such as the chain rule and product rule. Higher-order derivatives describe how the rate of change itself changes. Partial derivatives extend the concept to functions of several variables and lead to gradients and directional derivatives.

In finance, a derivative is a contract whose value is derived from an underlying asset, such as a stock, bond, commodity, currency, or index. Common examples include forwards, futures, options, and swaps. Derivatives are used for hedging risk, speculating on price movements, or obtaining exposure to assets without owning them. Pricing relies on models of future price behavior, and markets for derivatives are subject to regulation and risk, including leverage and counterparty risk.

In other areas, derivative can refer to chemical derivatives—compounds obtained from another substance by a defined reaction—or to linguistic derivatives, where new words are formed from base forms by affixation or a similar process.