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 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).

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 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 its 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).