I am having a hard time while trying to fully understand Hadamard differentiability.

I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und Evolutionsprobleme", Lecture notes, Technische Universität München, 2014):

$F: V \rightarrow W$ is Hadamard differentiable in $u\in V$ iff

$$\lim_{t\searrow 0} \frac{F(u+th+r(t))-F(u)}{t}$$ exists for all $h\in V$ and every $r:(0,\infty)\rightarrow V$ with $\lim_{t\searrow 0}\frac{r(t)}{t}=0$.

My aim is to prove the chain rule formula for Hadamard derivable functions and highlight the contrast with respect to non-Hadamard-differentiable functions, for which the chain rule does not hold in general. To do so I am looking for an example of a simple function, for which the directional derivative exists but Hadamard derivative does not exist. I couldn't find any such example: can someone of you help or maybe provide a good source?

Thank you so much!