Inverse functions and differentiation

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 as , where if and only if .


Example for arbitrary :

Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:

This relation is obtained by differentiating the equation in terms of x and applying the chain rule, yielding that:

considering that the derivative of x with respect to x is 1.

Writing explicitly the dependence of y on x, and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation):


This formula holds in general whenever is continuous and injective on an interval I, with being differentiable at () and where. The same formula is also equivalent to the expression

where denotes the unary derivative operator (on the space of functions) and denotes function composition.

Geometrically, a function and inverse function have graphs that are reflections, in the line . This reflection operation turns the gradient of any line into its reciprocal.[1]

Assuming that has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula.