Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$.
I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le n$).
I totally understand the proof when we replace $(H^1(\Omega))'$ by $H^{-1} = (H^1_0(\Omega))'$, but for the proof of $H^{-1}$ we crucially use the definition of the derivative of a distribution, to say that $$ < \partial_{x_i} u , \varphi > = - <u, \partial_{x_i} \varphi> $$ for all $\varphi \in \mathcal{C}^\infty_c(\Omega)$ and then we extend the linear form. But here, we want to take $\varphi \in \mathcal{C}^\infty(\bar{\Omega})$, so how what is the first step? (a reference and/or a complete proof will be perfect)