For $u \in C^1(\Omega)$ and $\phi \in C_0^{\infty}(\Omega)$ one has $$ \begin{split} \int_{\Omega}\nabla|u| \phi dx &=-\int_{\Omega}|u|\nabla \phi dx\\ &=-\int_{[u>0]}u\nabla \phi dx +\int_{[u<0]}u\nabla \phi dx\\ &=\int_{[u>0]}\nabla u \phi dx-\int_{[u<0]} \nabla u \phi dx \\&=\int_{\Omega}\operatorname{sign}(u) \nabla u \phi dx. \end{split} $$ For $u \in W^{1,1}({\Omega})$ set $$ \nabla|u|=\operatorname{sign}(u) \nabla u, $$ which is well-defined, since $\operatorname{sign}(u) \in L^{\infty}(\Omega)$. If $C^1(\Omega)\ni u_m \rightarrow u \in W^{1,1}(\Omega)$ one finds $$ \begin{split} \bigg|\int_{\Omega}\operatorname{sign}(u) \nabla u \phi dx &-\int_{\Omega}\operatorname{sign}(u_m) \nabla u_m \phi dx\bigg| \\ &\leq \bigg|\int_{\Omega}(\operatorname{sign}(u)-\operatorname{sign}(u_m)) \nabla u \phi dx\bigg|+\bigg|\int_{\Omega}(\operatorname{sign}(u_m) (\nabla u-\nabla u_m) \phi dx\bigg|\\ &\leq \int_{\Omega}|\operatorname{sign}(u)-\operatorname{sign}(u_m)| |\nabla u ||\phi| dx+\int_{\Omega}|\nabla u- \nabla u_m||\phi| dx \end{split} $$ this goes to zero for m to infinite. This confirms $|u|$ having a weak derivative $\operatorname{sign}(u)\nabla u$.
Question. I only know this definition with $\bigg|\dfrac{\partial u}{\partial x_i}\bigg|$: why is $\nabla$ used ?