Calculus of a Single Variable, Chapter 5, 5.6, Section 5.6, Problem 43

The given function: f(x) =arctan(e^x) is in a form of inverse trigonometric function.
It can be evaluated using the derivative formula for inverse of tangent function:
d/(dx)arctan(u) = ((du)/(dx))/(1+x^2) .
We let u = e^x then (du)/(dx)= d/(dx) (e^x)= e^x .
Applying the the formula, we get:
f'(x)= d/(dx) arctan(e^x)
=e^x/(1 +(e^x)^2)
Using the law of exponent: (x^n)^m=x^(n*m) , we may simplify the part:
(e^x)^2 = e^((x*2)) = e^(2x)
The derivative of the function f(x) = arctan(e^x) becomes:
f'(x)= e^x/(1 +e^(2x)) as the Final Answer.