Consider a locally integrable function $u$ on $\mathbb{R}^n$ (in $C^\infty$, for argument's sake). Then the function $f:[0,\infty)\times \mathbb{R}^n\rightarrow\mathbb{R}$ given by \begin{equation} f(r,x) = \int_{B_r(x)} u(y)\,dy = \int_{B_r(0)} u(x+y)\,dy \end{equation} is well-defined, with \begin{equation}\tag{1} \frac{\partial f}{\partial x_i}(r,x) = \int_{B_r(0)}\frac{\partial u}{\partial x_i}(x+y)\,dy = \int_{B_r(x)} \frac{\partial u}{\partial x_i}(y)\,dy = \int_{\partial B_r(x)} u(z)\nu^i(z)\,dS(z), \end{equation} by differentiation under the integral sign followed by the divergence theorem. Here, $\nu^i(z)$ is the $i$'th component of the outward pointing normal vector at $z\in\partial B_r(x)$.
I am interested in obtaining an analogue for $(1)$ for a function $u\in W_{\mathrm{loc}}^{1,p}(\mathbb{R}^n)$, meaning $u$ and its weak first derivatives belong to $L^p(K)$ for any compact $K\subset\mathbb{R}^n$. Of course, the final equality in $(1)$ cannot be obtained without further regularity assumptions on $u$, but do we still have the identity \begin{equation} \frac{\partial f}{\partial x_i}(r,x)= \int_{B_r(x)} \frac{\partial u}{\partial x_i}(y)\,dy, \end{equation} with weak derivatives? I don't believe the 'Measure Theory Statement' of this wiki article applies, since the second point only holds a.e. A reference to a book addressing this sort of stuff would also be welcome. Thank you!