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
$$ \frac{\partial}{\partial r}\int_{\partial B(0,r)}\frac{f(x)}{r}dx. $$