a derivative is the rate of change of a quantity. A derivative is an instantaneous rate of change: it is calculated at a specific instant rather than as an average over time. The process of finding a derivative is called differentiation. The reverse process is integration. The two processes are the central concepts of calculus and the relationship between them is the fundamental theorem of calculus.
This article assumes an understanding of algebra, analytic geometry, and the limit.
For a real-valued function of a single real variable, the derivative at a point equals the slope of the line tangent to the graph of the function at that point. Derivatives can be used to characterize many properties of a function, including
whether and at what rate the function is increasing or decreasing for a fixed value of the input to the function
whether and where the function has maximum or minimum values
The concept of a derivative can be extended to functions of more than one variable (see multivariable calculus), to functions of complex variables (see complex analysis) and to many other cases.
Differentiation has many applications throughout all numerate disciplines. For example, in physics, the derivative of the position of a moving body is its velocity and the derivative of the velocity is the acceleration.