How would I take this derivative? I am not really sure how I would take the following derivative:
$\frac{\partial}{\partial r}\left( F(r) \right) = \frac{\partial}{\partial r}\left( \int_{0}^{2 \pi} f(r,\theta) d\theta \right)$
Would it maybe just be:
$\frac{d}{dr}\left( F(r) \right)  = \left( \int_{0}^{2 \pi} \frac{\partial f(r,\theta)}{\partial r} d\theta \right)$
Is this right?
 A: This is correct. Because the boundaries of the integral do not depend on $r$, the derivative can be simply moved inside the integral.
If the bounds of integration did depend on $r$, extra terms from the boundaries would have to be included - look up "differentiation under the integral" if you are interested. :)
A: Your suggestion is correct, but only subject to certain conditions which your function $f$ will probably meet in practice.
This is called differentiation under the integral sign (link to Wikipedia).
The topic (by which I mean the question of "when is this interchange of symbols valid?") is covered in, among other books, Apostol's Mathematical Analysis (listed in the index) and Buck's Advanced Calculus (listed in the index under "Differentiation of functions defined by integrals").
I find this pdf by K. Conrad to be a good introduction to the actual use of the technique to solve problems, and is probably more in-depth than what you're looking for at the moment, but does have a precise statement of the relevant theorem towards the end.
