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

, then the inverse function rule is, in Lagrange's notation, This formula holds in general whenever

is continuous and injective on an interval I, with

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.

be an invertible (bijective) function, let

Since f is a bijective function,

is an invertible function, we know that

The inverse function rule can be obtained by taking the derivative of this equation.

The right side is equal to 1 and the chain rule can be applied to the left side: Rearranging then gives Rather than using

as the variable, we can rewrite this equation using

, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.

This can be shown using the previous notation

Then we have: By induction, we can generalize this result for any integer

, the nth derivative of f(x), and

: The chain rule given above is obtained by differentiating the identity

One can continue the same process for higher derivatives.

Differentiating the identity twice with respect to x, one obtains that is simplified further by the chain rule as Replacing the first derivative, using the identity obtained earlier, we get Similarly for the third derivative: or using the formula for the second derivative, These formulas are generalized by the Faà di Bruno's formula.

These formulas can also be written using Lagrange's notation.

If f and g are inverses, then so that which agrees with the direct calculation.