I would like to make this statement rigorous:
I have a smooth function $E:(a,b)^3\rightarrow \mathbb{R}$ and
I would like to prove that $\int_{\{(a,b); E(a,b)<k\}}1 dx$ is differentiable as a function of $k$ if $E^{-1}(k)$ contains only regular points, then:
$\frac{d}{dk}\left(\int_{\{(a,b); E(a,b)<k\}}1 dx\right) = \int_{E^{-1}(k)} \frac{1}{|\nabla E(x)|} dS(x).$
If you can only show it under weaker /additional assumptions, then this is also perfectly fine.
This is a sort of higher-dimensional variant of $\frac{d}{dx} \int_a^x F(s)ds= F(x).$