Calculus: Early Transcendentals, Chapter 4, 4.6, Section 4.6, Problem 22

We will solve the problem using Wolfram Mathematica (you can also use Wolfram Alpha which is free).
Graph of
Plot[1/(1 + E^Tan[x]), {x, - Pi/2, Pi/2}]
Plot[1/(1 + E^Tan[x]), {x, -6 Pi, 6 Pi}]
First line of code is used to make the first image and second to make the second image. The vertical lines in the second image are actually asymptotes.
Finding
Simplify[D[1/(1 + E^Tan[x]), x]]
In the code line above command Simplify is there to simplify the result obtained by differentiation.
The result is f'(x)=-(e^tan(x) sec^2(x))/(1 + e^tan(x))^2