I am looking to see if the following statement is true:
Let $\mathbb{V}$ be a Banach space. If $g:[0,\infty)\to \mathbb{V}$ is nonconvex and differentiable, and $f:\mathbb{V}\to \mathbb{R}$ is convex, lower-semicontinuous and proper, then $\frac{d}{dt}(f\circ g)(t) = p\frac{d}{dt}g(t)$ for some $p\in \partial f(g(t))$.
Can you help me show that this statement is true or false?
Considerations
If someone can prove true the statement but only with $\frac{d}{dt}$ meaning the right-derivative, then I am happy too.
This question is similar to two statements about convex compositions, namely
Let $\mathbb{V}$ be a Banach space. If $g:\mathbb{V}\to \mathbb{R}$ is convex, and $f:\mathbb{R}\to \mathbb{R}$ is convex, increasing and differentiable, then $\partial (f\circ g)(t) = f'(g(t))\partial g(t)$.
and
Let $\mathbb{V},\mathbb{U}$ be Banach spaces. If $K:\mathbb{U}\to\mathbb{V}$ is a linear bounded operator, $f:\mathbb{V}\to\mathbb{R}$ is proper, convex and lower-semicontinuous and there exists a $x_0$ such that $Kx_0\in int(dom(f))$, then $\partial(f\circ K)(x) = K^\ast\partial f(Kx)$ for all $x\in dom(f\circ K)$.
These are theorem 4.19 and theorem 4.17 of the book 'Introduction to Nonsmooth Analysis and Optimization' by Clason and Valkonen. This book has also been brought up in other questions, like this one. It is worth noting here that in the first statement the differentiability of $f$ is not needed (see remark 4.20 of that book).
Using the definition of the right-derivative and the subgradient, one can show that $$ \frac{d}{dt} (f\circ g)(t) \geq p\frac{d}{dt}g(t) $$ for all $p\in \partial f(g(t))$.
