Let $\Omega$ be a open set of $\mathbb{R}^N$, $u: \Omega \to \mathbb{R}$ a function, $ u \in C^2(\Omega)$. Given $x_0 \in \Omega$, let $r > 0$ be such that $B_r(x_0) \subset \subset \Omega$. For $0< s\leq r$, consider the following function:
$\varphi(s) = \dfrac{1}{n\omega_ns^{n-1}}\displaystyle\int_{\partial B_s(x_0)} u(x) \hspace{0.1cm}dS_x$, where $\omega_n = |B_1(0)|$.
Making the change of variable $x = x_0 + sz$, $z \in \partial B_1(0)$, we conclude easily that
$\varphi(s) = \dfrac{1}{n\omega_n}\displaystyle\int_{\partial B_1(0)} u(x_0 + sz)\hspace{0.1cm} dS_z$.
Ok. I'm having trouble justifying why the following equality is valid:
$\varphi'(s) = \dfrac{1}{n\omega_n}\displaystyle\int_{\partial B_1(0)} \nabla u(x_0 + sz)\cdot z \hspace{0.1cm}dS_z$.
My question then is: How to justify this equality? This looks like a Leibniz rule of calculus, but I feel unsure about applying it in this context.