Question 1:

Is the derivative of $\displaystyle f(x)=\frac{tanx}{1-2x}$

$\displaystyle f'(x)=\frac{(1-2x)sec^2x+2tanx}{(1-2x)^2}$ ?

Mr F says: Yes. And it's perfectly fine written like that.
Question 2:

If you were going to simplify this result, how would you do it?

My Gut is telling me to divide each term in the numerator by (1-2x).

Mr F says: That will give you something more complicated, not simpler, because the second term in the numerator does not contain a factor of (1 - 2x).
I don't think that it really matters.

I guess my question here is:

Does a a fraction look more simple than two seperate terms?

Mr. Fantastic would say, "Poppycock!"