Derivatives of Inverse Functions

From Department of Mathematics at UTSA
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Rule:


Example for arbitrary :

In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of . The inverse of is denoted as , where if and only if .

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 and applying the chain rule, yielding that:

considering that the derivative of with respect to is 1.

Writing explicitly the dependence of on , 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 , 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.

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.


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