$v(t) = \int_{t_0}^t u(\tau) \mathop{d\tau} $ is weakly differentiable

Question

Let E subset of R be an open interval and $p \in [1, \infty)$. We should show the following:

For $ u \in L^p(E)$ and $t_0 \in E$ show that

$v(t) = \int_{t_0}^t u(\tau) \mathop{d \tau} $ is weakly differentiable and that $v'=u$ almost everywhere.

Answer 1

I actually have this answer, but would appreciate an explanation as there are some steps I do not understand. Thank you!