# How do you differentiate #arctan(x^2)#?

##### 2 Answers

#### Explanation:

The derivative of the arctangent function is:

#d/dxarctan(x)=1/(1+x^2)#

So, when applying the chain rule, this becomes

#d/dxarctan(f(x))=1/(1+(f(x))^2)*f'(x)#

So, for

#d/dxarctan(x^2)=1/(1+(x^2)^2)*d/dx(x^2)#

#=(2x)/(1+x^4)#

#### Explanation:

Let

Then

#sec^2(y)*dy/dx=2x#

Dividing both sides by

#dy/dx=cos^2(y)*2x#

#dy/dx=cos^2(arctan(x^2))*2x#

Note that

If

Since cosine is the adjacent side,

Therefore

#dy/dx=(1/sqrt(1+x^4))^2*2x=1/(1+x^4)*2x=(2x)/(1+x^4)#