There are related questions and answers on StackExchange but I cannot find a general formula for the following types of (time) derivatives of integrals with respect to changing level and upper contour sets.
Let $x$ be a $d$-dimensional vector and $t$ be a scalar, and let $f(x,t)$ be a function mapping from $\mathbb{R}^n\times\mathbb{R}$ to $\mathbb{R}$. Assume that $0$ is a regular point of $f(\cdot, t)$ for every $t$.
(1) Define $$\phi(t):=\int_{\{x: f(x,t) \geq 0\}} w(x) dx.$$ How to derive a formula for $\frac{d}{dt} \phi(t)$?
(2) Define $$\psi(t):=\int_{\{x:f(x,t) = 0\}} w(x) d\mathcal{H}^{d-1}(x),$$ where $\mathcal{H}^{d-1}$ denotes the $(d-1)$ dimensional Hausdorff measure. How to derive a formula for $\frac{d}{dt} \psi(t)$?