Let $R_t$ be a family of compact, simply connected regions in the plane defined by

$R_t = \{x\in\mathbb{R}^2 : h(x) \leq t\}$

for all $t$, where $h(x)$ is some nicely behaved smooth function. Suppose $f(x)$ is a probability density on $\mathbb{R}^2$ and define

$M(t) = \iint_{R_t} f(x) dA$

for all $t$. Is it true that

$\frac{dM}{dt}|_{t=t_0} = \int_{\partial R_{t_0}} f(x) /\|\nabla h\| ds$

where $ds$ denotes integration with respect to arc length? If not, what is the right expression for $\frac{dM}{dt}$? I assume this is some well-known first-year calculus-type problem but I can't find it stated in any context (it may very well be a common homework problem though I've not seen it).