Inverse function rule

In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of is denoted as , where if and only if , then the inverse function rule is, in Lagrange's notation,

.
Rule:


Example for arbitrary :

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.

The inverse function rule may also be expressed in Leibniz's notation. As that notation suggests,

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.


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