Let us consider a convex function $f : \mathbb{R}^n \rightarrow \mathbb{R}$. Let us consider that is composed with an absolutely continuous function of a real variable $t$, $ x : \mathbb{R} \rightarrow \mathbb{R}^n, t \mapsto x(t)$.
The question is : what is the derivative of $f(x(t))$ with respect to $t$? can we write
$$ \frac{d}{dt} f(x(t)) = \xi \frac{d}{dt}x(t) \text{ with } \xi \in \partial f(x(t)) \text{ almost everywhere}$$
For instance, if $f(x) = |x|$, we have
$$ \partial |x| = \mbox{sign}(x) = \begin{cases}-1, x <0 \\ 1, x>0 \\ [-1,1], x=0\end{cases}$$
Can we write
$$ \frac{d}{dt} |x(t)| = \xi \frac{d}{dt}x(t) \text{ with } \xi \in \mbox{sign}(x(t)) $$