derivat

Derivat is a term used in several languages to denote something derived from another. In mathematics and related fields it most often refers to the derivative, the rate at which a quantity changes with respect to a variable. In chemistry and biochemistry, a derivative is a compound formed from another by modification of its structure. In finance, a derivative is a contract whose value is based on an underlying asset.

In calculus, the derivative of a function f at a point x is the limit as h approaches zero of [f(x+h) − f(x)]/h, provided the limit exists. It measures the instantaneous rate of change and equals the slope of the tangent to the graph of f at x. Notation includes f′(x), dy/dx, and Df(x).

Differentiation rules, such as the power rule, product rule, and chain rule, enable finding derivatives of a wide range of functions. For functions of several variables, partial derivatives ∂f/∂x, the gradient, and higher derivatives like the Hessian describe rates of change in multiple directions.

In physics, derivatives describe velocity as the time derivative of position and acceleration as the derivative of velocity, linking calculus to motion.

In chemistry, a derivative is a compound derived from a parent by substitution or modification of its structure; derivatization is used to improve properties or to enable detection and analysis in analytical methods.

In finance, derivatives are contracts whose value derives from underlying assets such as stocks, bonds, indices, or interest rates. Common forms include options, futures, forwards, and swaps; they are used for hedging, leverage, and risk management, though they carry model and market risks.