$\| X'(t) \| = (\| X(t) \|)'$ 
Let $X : \mathbb{R} \to \mathbb{R}^n$ be a $C^1$ function. Let $\| .\|$ be the norm : $\| v \| = \max_{1 \leq i \leq N} \mid v_i \mid$. Then is it true that : 
  $$\| X'(t) \|  = (\| X(t) \|)'$$ ?

I am wondering if in general if I have any function $f : \mathbb{R}^n \to \mathbb{R}^p$ and a norm $N$ on a : $\mathbb{R}^p$ then is it always possible to invert the norm and the differential operator or the norm and in the integral?
Thank you. 

For $n=1$ the identity function $X(x)=x$ is a $C^{1}$ function. In this case $|X(x)|$ is not even differentiable at $0$. 
