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 in terms of the derivative of. More precisely, if the inverse of is denoted as, where if and only if, then the inverse function rule is, 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 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.
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 and applying the chain rule, yielding that:
considering that the derivative of with respect to is 1.

Derivation

Let be an invertible function, let be in the domain of, and let Let So, Differentiating this equation with respect to, and using the chain rule, one gets
That is,
or

Examples

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

    Additional properties

  • Integrating this relationship gives
  • Another very interesting and useful property is the following:
  • The inverse of the derivative of f is also of interest, as it is used in showing the convexity of the Legendre transform.
Let then we have, assuming :This can be shown using the previous notation. Then we have:
By induction, we can generalize this result for any integer, with , the nth derivative of f, and , assuming :

Higher order derivatives

The chain rule given above is obtained by differentiating the identity with respect to, where. 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
These formulas can also be written using Lagrange's notation:
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:
From this expression, one can also derive the th-integration of inverse function with base-point using Cauchy formula for repeated integration whenever :

Example

  • has the inverse. Using the formula for the second derivative of the inverse function,
so that
which agrees with the direct calculation.