Derivative (calculus)

A derivative is a measure in Calculus of how functions change based on how their input values change; otherwise known as 'the rate of change' (second derivatives thus give the rate of change of change, etc). For example, velocity is obtained from taking the first derivative of a position function with respect to time. Acceleration can be obtained either by taking the derivative of a velocity function with respect to time or by taking the time derivative of a position function twice. Note that when using this method, taking the derivative the first time yields the velocity function, and taking the derivative the second time yields the acceleration function.

The terms speed and velocity are often used interchangeably, which is incorrect. Speed is a Scalar quantity which refers to the magnitude, or size, of the velocity vector. Velocity has a direction, while speed does not.

Given a graph of a real curve, the derivative at a specific point will equal the slope of the line tangent to that point. 

To calculate the derivative of a function, one must use techniques from the differential branch of calculus. This branch of calculus is related to the integral branch by the first fundamental theorem of calculus: differentiation (the process of finding a derivative) is the reverse process of integration (the process of finding an integral).

In mathematics, derivatives are helpful in determining the maxima and minima of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called critical points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the second derivative test. The second derivative is used to determine the concavity, or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of inflection, meaning that it is a point where the concavity could be changing.