Compute the following norm derivatives

Question

I was wondering if anyone can explain me how to compute the derivatives of the following norms:

1. $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$
2. $\bigtriangledown ||u(x)||^2_2$ where $u: \mathbb{R^n} \rightarrow \mathbb{R^n} $ is sufficiently smooth, and
3. $\Delta||x||_2:=div(\bigtriangledown||x||_2):=\bigtriangledown \cdot (\bigtriangledown||x||_2)$ for $x\in\mathbb{R^n}$

Thank you very much in advance!

Answer 1

(1) is a composition of the next functions: $$\mkern-1em j_p:\Bbb R\longrightarrow \Bbb R^n$$ $$s\longmapsto sp$$

$$\mkern-1em T_z:\Bbb R^n\longrightarrow \Bbb R^n$$ $$v\longmapsto x+v$$

$$\mkern-1em N_q:\Bbb R^n\longrightarrow \Bbb R$$ $$v\longmapsto \|v\|_q$$

$$\mkern-1em S:\Bbb R\longrightarrow \Bbb R$$ $$t\longmapsto t^2$$

Compute the derivatives of each function and use the chain rule.