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Update: Solution for Example 1
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*It seems the right apparatus for studying such problems is *Mallavian calculus*. I've only come to touch with this tool a few hours ago, and its quite possible I don't know what I'm talking about below.*

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So, let $G = (g_1,\ldots,g_d)$ be a standard Gaussian random vector in $\mathbb R^d$ and consider the Gaussian process defined by $Z(a) := \langle G,a\rangle = \sum_j a_j g_j$, for any $a \in \mathbb R^d$. The random variable "$f(x)$, $x \sim N(0,I_d)$" in the question can be written as $X := Z(w)-c+\sum_{j=1}^d w_j m_j$. Let $S(X):=\delta(DX/\|DX\|^2)$, where $D$ (resp. $\delta$) is the *Mallavian derivative* (resp. *Skorohod integral*) operator. A simply computation gives $DX = \sigma \sum_j w_j e_j = \sigma w$, and so $S(X) = \sum_j w_j Z_j / (\sigma \|w\|^2) = (\sigma\|w\|)^{-1}Z(w)$.

Therefore, thanks to **Proposition 2.1.1** of [Introduction to Mallavian Calculus][1], one obtains

$$
s_f'(t) = \mathbb E_z[1_{X \le t} S(X)] = \frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^{(t-f(m))/\sigma}ze^{-z^2/2}dz = \varphi(\frac{t-f(m)}{\sigma}),
$$
where $\varphi$ is the standard Gaussian pdf.


  [1]: https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/kunze/malliavin/Malliavin_skript.pdf