Continuously differentiable functions are weakly differentiable

Question

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial x_i}d\lambda^n=-\int_\Omega u\frac{\partial \psi}{\partial x_i}d\lambda^n\;\;\;\text{for all }\psi\in C_c^1(\Omega)\;.$$ It doesn't seem to be elementary provable. Do we need to approximate $u$ or $\partial_iu$ in some way?

Answer 1

The proof that you are looking for can be found in the page 258 of the Evans book: