This is the proof I saw in my text.
I can understand all the algebraic manipulations in the proof. But, what I am not sure about is the deriative of the function Φ(r). As far as I know, the average value of a function u(x) over the surface of a ball is $\frac1{s(r)}\int_{∂ ball}u(x) ds$, where s(r) is the surface area of the ball that depends on r. In the text, I saw differentiation with respect to r. Now,here comes my question. If we differentiate with respect to r, with the function $\frac1{s(r)}$ standing outside of the integral, shouldn't we use the product rule? In the text, I only saw the intgral get differentiated as if the average value is obtained by dividing by a constant and not a function of r.
This derivative is calculated by first changing variables; intuitively, it's weird to differentiate an integral with respect to $r$, when the integral is over a region that depends on $r$. The change of variables in the first picture does this. It takes the $r$ out of the limits of the integral, so now we can just pull the derivative inside. This is also why we don't need the product rule; changing variables as such makes the fraction in front of the integral a constant (the volume of the $n$-dimensional unit sphere). To make this rigorous can be done using $n$-dimensional "polar" coordinates.