I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE.  When coming across [this post][1] I could not help but wonder the following.  

Let $H$ be the Reproducing-Kernel Hilbert space obtained by completing the set of all $C^1(\mathbb{R}^n,\mathbb{R}^m)$ with finite norm finite:
$$
\|f(x)\|_H:= \|f(0)\|_{\mathbb{R}^m} + \int_{x \in \mathbb{R}^n} \|(\nabla f)(x)\|_{\mathbb{R}^m} e^{-\|x\|} dx.
$$

If $w:C^1(\mathbb{R}^n,\mathbb{R}^m)\times \mathbb{R}^n\rightarrow [0,\infty)$ is the functional
$$
(f,x) \mapsto \|f(x)\|_{\mathbb{R}^m},
$$
what is it's Fréchet derivative?  Thinking analogously to the linked post and appealing to the chain-rule for Fréchet derivatives, I would guess it is 
$$
Dw(f,x) (g,y)= \frac1{\|g(x)+J_f(y)\|}\left(g(x) + (J_f)(y)\right).
$$
However, I don't know how to show more than this (if even it is a correct ansatz).  


  [1]: https://math.stackexchange.com/questions/2463116/frechet-derivative-of-evaluation-function