# Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$\frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y),$$ where $dS$ is the Lebesgue surface measure on $\mathbb{R}^{d-1}$. For $d=3$, it is a particular case of the Reynolds transport theorem, however, how is it called or where was it formulated for other dimensions?

Clearly, the question is not about the proof, it is more or less straightforward, but about any reference only.

By the way, what is about the 'maximum' class of functions, for which this formula does hold? $L^1_{\mathrm{loc}}(\mathbb{R}^d)$ ?