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.
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.
