In mathematics, a derivative is a fundamental tool of calculus that shows the sensitivity of change of a functions output with respect to the input. It is the rate of change of a quantity y with respect to another quantity x. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. The concept of a derivative can be extended to many other settings, such as complex functions of complex variables, maps between infinite dimensional vector spaces, and more.

The process of finding a derivative is called differentiation. The derivative of a function is the measure of the rate at which the value of y changes with respect to the change of the variable x. The derivative of a function f(x) in math is denoted by f(x) and can be contextually interpreted as the slope of the tangent drawn to that curve at that point. It also represents the instantaneous rate of change at a point on the function. The derivative of a function can be used to find the maximum and minimum values of the function, the concavity of the function, and more.

Different notations are used to represent derivatives, such as Leibnizs, Lagranges, and Newtons notations. The derivative of a function can be calculated using the limit definition of the derivative, which involves taking the limit of the difference quotient as the change in the input variable approaches zero. The derivative of a function can also be found using differentiation rules, which are shortcuts for finding the derivative of more complex functions.