In mathematics, the inverse of a function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see inverse function for a formal and detailed definition). The inverse of f is denoted f ^{- 1}. The statements y=f(x) and x=f^-1(y) are equivalent.

Differentiation in calculus is the process of obtaining a derivative. The derivative of a function gives the slope at any point.

denotes the derivative of the function y = f(x) with respect to x.

denotes the derivative of the function x = f(y) with respect to y.

The two derivatives are, as the Leibniz notation suggests, reciprocal, that is

This is a direct consequence of the chain rule, since

and the derivative of x with respect to x is 1. Geometrically, a function and inverse function have graphs that are reflections, in the line y=x. This reflection operation turns the gradient of any line into its reciprocal.

Examples

• y = x² (for positive x) has inverse .

At x=0, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.

• y = e^x has inverse x = ln(y) (for positive y).

Additional properties

• Integrating this relationship gives

This is only useful if the integral exists. In particular we need f'(x) to be non-zero across the range of integration.

It follows that functions with continuous derivative have inverses in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.

Related topics

calculus, inverse functions, chain rule, inverse function theorem, implicit function theorem.