In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function in terms of the derivative of , then the inverse function rule is, in Lagrange's notation,

is continuous and injective on an interval . The same formula is also equivalent to the expression

denotes function composition.

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

has an inverse in a neighbourhood 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,

in terms of and applying the chain rule, yielding that:

considering that the derivative of with respect to is 1.

Derivation

Differentiating this equation with respect to , and using the chain rule, one gets

That is,

or

Examples

• (for positive .

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

• (for real )

Additional properties

• Integrating this relationship gives

• to be non-zero across the range of integration.

  It follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.

• Another very interesting and useful property is the following:

• .

• The inverse of the derivative of f(x) is also of interest, as it is used in showing the convexity of the Legendre transform.

. Then we have:

1. Therefore:

:

Higher order derivatives

The chain rule with respect to . One can continue the same process for higher derivatives. Differentiating the identity twice with respect to , one obtains

that is simplified further by the chain rule as

Replacing the first derivative, using the identity obtained earlier, we get

which implies

Similarly for the third derivative we have

Using the formula for the second derivative, we get

which implies

In general, higher order derivatives of an inverse function can be expressed with Faà di Bruno's formula. Alternatively, the th derivative can be written succinctly as an application of a differential operator:

after:

One can also derive the th-integration of inverse function with base-point using Cauchy formula for repeated integration:

Example

• . Using the formula for the second derivative of the inverse function,

so that

which agrees with the direct calculation.

See also

References

Citations

Sources