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, speed is a first derivative, obtained from distance and time, whereas acceleration is a second derivative, obtained from the first derivative, speed, and time.

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.