Domain and Derivative of 2[arctan(e^x)]?

[fk378]

Homework Statement

Find the domain and the first derivative of 2[arctan(e^x)]

Homework Equations

d/dx arctan(x)= 1/(1+x^2)

The Attempt at a Solution

I'm not sure about the domain...

For the derivative:
d/dx 2[arctan(e^x)] = 2 [1/(1+e^x)^2] (e^x)

But my teacher had the same answer excluding the e^x part. Isn't there supposed to be the e^x there since it is the derivative of the inside function?

 

[Dick]

Yes, your teacher is wrong. But (1+e^x)^2 isn't the denominator. What should it be? To answer the domain question you first have to figure out the domain of arctan.

 

[fk378]

Oh right, the answer should be 2 [1/1+(e^x)^2]
The domain of arctan is (infinity, infinity)? (Since the function covers the entire graph)

 

[Dick]

Right, so put that together with the domain of e^x. Are there any values of x where the function isn't defined?

 

[fk378]

Well the graph of e^x looks like it's not defined anywhere below y=0. So would that mean the domain of e^x is x>0?

 

[Dick]

You are confusing the domain and the range of e^x. For what values of x (not y!) is e^x defined.

 

[fk378]

Oh, the domain is all reals then...so the domain of arctan(e^x) is all reals!

 

[Dick]

Yep. I agree.