Question in the proof of Rellich-Kondrachov theorem

Question

I'm reading Chapter 5 of Evans' book 《Partial differential equations. 2nd edition》 to understand some basic facts about Sobolev spaces and I have some questions in his proof of Rellich-Kondrachov theoremProof of Rellich-Kondrachov theorem

My question is why the following equality is true?

$u_m(x-\epsilon y)-u_m(x)=\int_0^1\frac d {dt}(u_m(x-\epsilon ty))dt$, where $u_m\in W^{1,p}(\mathbb{R}^n)$ with compact support in some bounded open set $V\subset\mathbb{R}^n$.

In conclusion, my questions are following:

1. If $t\in I=(0,1)$ and $u\in W^{1,p}(\mathbb{R}^n)$ with $supp(u)\subset V$, where $V$ is a bounded open set of $\mathbb{R}^n$. Do we have the following: For a.e. $x\in\mathbb{R}^n$, the function $t\mapsto u(tx)$ belongs to $W^{1,p}(I)$ ? (Maybe this is equivalent to $t\mapsto u(tx)$ absolutely continuous on the interval $I$)

   This proposition is trivial when $u\in C_c^{\infty}(\mathbb{R}^n)$. I want to use the approximation arguement but failed.

2. If 1 is true, do we have the following: $\frac{d}{dt}u(tx)=Du(tx) \cdot x$, where ''$\cdot$'' is the inner product in $\mathbb{R}^n$ ?

   Any advice is helpful, thanks a lot.

Edition: I omitted the condition of assuming $u_m$ smooth at first. There is no problem about the equality. But can we use this proof to the case "$u_m\in W^{1,p}$" directly, without assuming $u_m$ smooth first?