Inverse functions and differentiation

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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.