Inverse functions and differentiation

From Wacklepedia - The Free Encyclopedia

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.

Table of contents

1 Examples
2 Additional properties
3 Related Topics

Examples

(for positive ) has inverse .

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.