Derivatives of Inverse Functions

Inverse functions are functions that "reverse" each other.

We consider a function f (x), which is strictly monotonic on an interval (a, b). If there exists a point x₀ in this interval such that f '(x₀) ≠ 0, then the inverse function x = φ (y) is also differentiable at y₀ = f (x₀) and its derivative is given by

Let us prove this theorem (called the inverse function theorem).

Suppose that the variable $$ gets an increment $$ at the point $$ The corresponding increment of the variable $$ at the point $$ is denoted by $$, where $$ due to the strict monotonicity of $$. The ratio of the increments is written as

Suppose that $$. Then $$, since the inverse function $$ is continuous at $$. In the limit when $$, the right side of the relationship becomes

In this case, the left hand side also approaches a limit, which by definition is equal to the derivative of the inverse function:

Thus,

that is the derivative of the inverse function is the inverse of the derivative of the original function.

In the examples below, find the derivative of the function $$ using the derivative of the inverse function $$

Solved Problems

Solution.

We first determine the inverse function for the given function $$. To do this, we express the variable $$ in terms of $$

By the inverse function theorem, we can write:

Now we substitute $$ instead of $$ As a result, we obtain an expression for the derivative of the given function:

Solution.

The arcsine function is the inverse of the sine function. Therefore $$ $$ Then the derivative of $$ is

where $$

Solution.

The natural logarithm and the exponential function are mutually inverse functions. Therefore, $$, where $$, $$. The derivative of the natural logarithm is easy to calculate through the derivative of the exponential function:

Here we have used the logarithmic identity

Solution.

We first find the inverse function $$ for the given function $$ which is monotonically increasing for any $$. Express $$ in terms of $$

Now find the derivative $$

Solution.

The arccosine function is defined and monotonic on the interval $$. Consequently, the domain of the original function has the form:

Write the inverse function $$

Calculate the derivative of the original function through the derivative of the inverse function:

Note that the derivative is not defined at the boundary points $$ and $$ of the domain of the function $$

Solution.

This function is defined and monotonically increasing for $$. Therefore we can construct the inverse function on this interval. Express $$ in terms of $$

Now we define the derivative of the given function $$ using the inverse function theorem:

Substitute the expression for the original function instead of $$