Inverse Functions and Differentiation

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 . The statements y = f(x) and x = f −1(y) are equivalent.

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

This is a direct consequence of the chain rule, since

and the derivative of with respect to is 1.

Writing explicitly the dependence of on and the point at which the differentiation takes place and using Lagrange's notation, the formula for the derivative of the inverse becomes

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.

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.