I'm studying all of this via distance education, so sometimes it can take a little bit to get the idea to set in, so this'll probably be another silly question.

Where you've written

how did you get to this point?

The only way I can see it being done, is similar to the manner I got it there (chain rule), but with the fact that it was (e^w)^2 in the denominator that makes it stay as that expression (not squared). When compared to the numerator, which only had (e^w), this can simply be represented by 1 - almost like factorization.