# What is the derivative of # (arctan x)^3#?

##### 1 Answer

#### Explanation:

We are given

By the power rule, if

Unfortunately, we can't directly apply the power rule to find

Instead, the power rule can only find the derivative with respect to

However, by the chain rule, we can multiply both sides by

(Note: the left hand side cancels out:

#dy/cancel(d(arctan(x)))*cancel(d(arctan(x)))/dx=dy/dx# .)

Now, we just need to find

Consider the function

Now, if we differentiate both sides, we will get

We said previously that

Note:

#cos^2(arctan(x))# is simplified using the identity#tan^2(theta)+1=1/cos^2(theta)# (found by dividing both sides of#sin^2(theta)+cos^2(theta)=1# by#cos^2(theta)# and using the fact that#tan(theta)=sin(theta)/cos(theta)# ).

This identity can be arranged to#cos^2(theta)=1/(tan^2(theta)+1)# .

Now, we have

Since