How can one compute the following integral? $ \frac{\partial}{\partial r}\int_{\partial B(0,r)}f(r,x)dx. $ I know a similar integral $ \frac{\partial}{\partial r}\int_{B(0,r)}f(r)dx =\int_{\partial B(0,r)}f(r)dx+\int_{B(0,r)}\frac{\partial}{\partial r}f(r)dx. $ Maybe I don't fully understand the latter integral either. I would appreciate it if someone could kindly explain what is going on.
More specifically I want to compute something like
$ \frac{\partial}{\partial r}\int_{\partial B(0,r)}\frac{f(x)}{r}dx. $