Inverse functions and differentiation Guide, Meaning , Facts, Information and Description

In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of (see inverse function for a formal and detailed definition). The inverse of is denoted . 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  with respect to .

denotes the derivative of the function  with respect to .

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 with respect to 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.

Table of contents

1 Examples 2 Additional properties 3 Related Topics

Examples

• (for positive ) 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.

• has inverse (for positive ).

Additional properties

• Integrating this relationship gives

This is only useful if the integral exists. In particular we need 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.

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