##
Differentiating Inverse Functions

Inverse functions are very important in Mathematics as well as in
many applied areas of science. The most famous pair of functions
inverse to each other are the logarithmic and the exponential
functions. Other functions like the tangent and arctangent play
also a major role.

In any case, let *f*(*x*) be an invertible function, with
*f*^{-1}(*x*) as its inverse, that is

May be the easiest way to remember this is to write

If *f*(*x*) is differentiable on an interval *I*, one may wonder
whether *f*^{-1}(*x*) is also differentiable? The answer to this
question hinges on *f*'(*x*) being equal to 0 or not . Indeed, if
for any ,
then *f*^{-1}(*x*) is also
differentiable. Moreover we have

Using Leibniz's notation, the above formula becomes

which is easy to remember.
**Example.** We will see in the coming pages that the
logarithmic function
is the antiderivative of
for *x* > 0, with .
This
function has an inverse on
,
known to us as the
exponential function *e*^{x}. So this function is differentiable
and

But we have
,
hence

This is truly an amazing result: the derivative of *e*^{x} is the
function itself. This property of the exponential function has
many interesting applications.
**Example.** **Rational Powers.** For *x* > 0, and any
natural numbers *n* and *m*, we have

Let us first take care of the derivative of the function *n*^{th}root
*f*(*x*)=*x*^{1/n}, which is just the inverse function of *x*^{n}.
Clearly for *x* > 0, the derivative of *x*^{n} is not 0, so
*f*(*x*)=*x*^{1/n} is differentiable and

Easy algebraic calculations give

In other words, the formula
is also
valid for *r*=1/*n*, for
Back to our formula, to
differentiate the function *x*^{n/m} we will use the above
results combined with the chain rule. So we have

Again this means that the formula
is
valid even when *r* is a rational number. In particular combined
with the chain rule, we get

for any differentiable function *u*(*x*).
**Example.** We have

The following example discusses the above ideas for trigonometric
functions.

**Example.** Consider the tangent function
on the
interval
.
Then
has an
inverse function
known as the **arctangent
function**, which we prefer to denote by
.
Since

for any
,
then
is
differentiable. Moreover we have

or

It might be surprising that the transcendental function
has as its derivative a rational function. This will
be very useful once we try to integrate rational functions.
**Exercise 1.** Compute

**Answer.**

**Exercise 2.** Compute

**Answer.**

**
**

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