Partial derivative computation

Question

Reading the book Subharmonic Functions by Tibor Rado, I came across the fact that the function $$A_r(x,y;f)=\frac{1}{\pi r^2}\int_{0}^{2\pi}\int_{0}^{r}f(x+\rho\cos\phi,y+\rho\sin\phi)\rho \ \mathrm{d}\rho \ \mathrm{d}\phi,$$ where $f$ is a real valued continuous function in a domain $G$, has continuous derivatives of the first order in the portion of $G$ where it is defined. It is claimed that the four-step rule for differentiation leads "immediately" to the formula $$\frac{\partial A_r(x,y;f)}{\partial x}=\frac{1}{\pi r}\int_{0}^{2\pi}f(x+r\cos\phi,y+r\sin\phi)\cos\phi \ \mathrm{d}\phi.$$ The problem is that I still struggle to figure out why is this calculation immediate. I've tried some changes of variables but it didn't help at all. Any ideas would be appreciated.