How do you find the derivative of #sqrt[arctan(x)]#?

2 Answers
Sep 21, 2016

Answer:

#1/(2(x^2+1)sqrtarctan(x))#

Explanation:

#y=sqrtarctan(x)#

#y=(arctan(x))^(1/2)#

We will use the chain rule. The specific adaptation of the chain rule that we will be using here is where we use the power rule on the outside function:

#d/dxu^(1/2)=1/2u^(-1/2)(du)/dx#

So:

#dy/dx=1/2(arctan(x))^(-1/2)d/dxarctan(x)#

#dy/dx=1/(2sqrtarctan(x))d/dxarctan(x)#

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

#dy/dx=1/(2(x^2+1)sqrtarctan(x))#

Sep 21, 2016

Answer:

#1/(2(x^2+1)sqrtarctan(x))#

Explanation:

#y=sqrtarctan(x)#

#y^2=arctan(x)#

Differentiate both sides. The chain rule will be used on the left!

#2ydy/dx=1/(x^2+1)#

Since #2y=2sqrtarctan(x)#:

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

#dy/dx=1/(2(x^2+1)sqrtarctan(x))#