Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, for every $t \ge 0$, define $$ s_f(t) := \mu(f^{-1}((-\infty,t])) = \int_{f^{-1}((-\infty,t])}p(x)dx $$

Question. What is the derivative of $s_f$ w.r.t $t$ ?

As examples for the function $f$, one could consider

• Linear: $f(x) = w^\top x - c$, for some unit-vector $w \in \mathbb R^d$ and scalar $c \in \mathbb R$.
• Quadratic: $f_\pm(x) = \pm (1-\|x\|^2)$.

It seems I should be able to solve my problem in principle using Lemma 3.1 of Malliavin Calculus for non Gaussian differentiable measures and surface measures in Hilbert spaces. However, that paper is hard to parse for a non-expert like myself.