# Inverse functions and differentiation

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.

- 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