derivativák

A derivative, in mathematics, represents the instantaneous rate of change of a function with respect to one of its variables. It is a fundamental concept in calculus, describing how a function's output changes as its input changes infinitesimally. Geometrically, the derivative of a function at a particular point corresponds to the slope of the tangent line to the function's graph at that point.

The process of finding a derivative is called differentiation. For a function f(x), its derivative is often denoted as f'(x) or dy/dx. The formal definition of a derivative involves a limit: the derivative of f(x) is the limit of the difference quotient [f(x+h) - f(x)]/h as h approaches zero. This limit, if it exists, provides the precise instantaneous rate of change.

Derivatives have wide-ranging applications across various fields. In physics, they are used to describe velocity and acceleration. In economics, they help analyze marginal cost and marginal revenue. Engineers use derivatives for optimization problems and to model dynamic systems. Understanding derivatives is crucial for solving problems involving rates of change, optimization, and the behavior of functions.