# Homework Help: Inverse tan derivative

1. Oct 17, 2007

### fk378

1. The problem statement, all variables and given/known data
Find the domain and the first derivative of 2[arctan(e^x)]

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

3. 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?

2. Oct 17, 2007

### 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.

3. Oct 17, 2007

### 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)

4. Oct 17, 2007

### Dick

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

5. Oct 17, 2007

### 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?

6. Oct 17, 2007

### Dick

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

7. Oct 17, 2007

### fk378

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

8. Oct 17, 2007

### Dick

Yep. I agree.