What's theorem can we use to derive a function given by a integral?

Question

Let $U\subset\mathbb{R}^n$ be a open set and $u\in C^2(U)$ a harmonic function. Suppose the following function is well defined. $$\phi(r)=\int_{\partial B(0,1)}u(x+rz)\;dS(z)$$ I know that $$\phi'(r)=\int_{\partial B(0,1)}\frac{\partial u}{dr}(x+rz)\;dS(z)$$ Can someone tell me what theorem we can use to justify this passage and where can I find a proof for it?

Thanks.

Answer 1

I think you want Leibniz integral rule.

http://en.wikipedia.org/wiki/Leibniz_integral_rule

Information is also here:

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign#Higher_dimensions