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n calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity at which the vehicle is traveling. Conversely, the integral of the velocity over time is the change in the vehicle's position. This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function \ƒ at a is the limit The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration. his rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = \ƒ(x), where \ƒ denotes the function.