  ## Encyclopedia > Inverse functions and differentiation

Article Content

# Inverse functions and differentiation

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.

$\frac{dy}{dx}$ denotes the derivative of the function $y=f(x)$ with respect to $x$.

$\frac{dx}{dy}$ denotes the derivative of the function $x=f(y)$ with respect to $y$.

The two derivatives are, as the Leibnitz notation[?] suggests, reciprocal, that is

$\frac{dx}{dy}\,.\, \frac{dy}{dx} = 1$

This is a direct consequence of the chain rule, since

$\frac{dx}{dy}\,.\, \frac{dy}{dx} = \frac{dx}{dx}$

and the derivative of $x$ with respect to $x$ is 1.

• $y = x^2$ (for positive $x$) has inverse $x = \sqrt{y}$.

$\frac{dy}{dx} = 2x \mbox{ }\mbox{ }\mbox{ }\mbox{ }; \mbox{ }\mbox{ }\mbox{ }\mbox{ } \frac{dx}{dy} = \frac{1}{2\sqrt{y}}$

$\frac{dy}{dx}\,.\,\frac{dx}{dy} = 2x . \frac{1}{2\sqrt{y}} = \frac{2x}{2x} = 1$

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

$\frac{dy}{dx} = e^x \mbox{ }\mbox{ }\mbox{ }\mbox{ }; \mbox{ }\mbox{ }\mbox{ }\mbox{ } \frac{dx}{dy} = \frac{1}{y}$

$\frac{dy}{dx}\,.\,\frac{dx}{dy} = e^x . \frac{1}{y} = \frac{e^x}{e^x} = 1$

• Integrating this relationship gives

${f^{-1}}(y)=\int\frac{1}{f'(x)}\,.\,{dx} + c$

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Holtsville, New York ... with them, 70.4% are married couples living together, 9.9% have a female householder with no husband present, and 16.2% are non-families. 12.0% of all households are made ...  