Let $f\colon \Omega\to \R$ and $g=(g_1,\ldots,g_n)\colon \Omega\to \R^n$ be locally integrable functions defined on an open set $\Omega\subset \mathbf R^n$ We say that $g$ is the weak derivative of $f$ if the equality $$ \int_\Omega f \frac{\partial \phi}{\partial x_i} = - \int_\Omega g_i \phi $$ holds true for all functions $\phi\in\mathcal C^\infty_c(\Omega)$ (smooth functions with compact support in $\Omega$ and for all $i=1,\ldots, n$ Notice that the integrals involved are well defined since $\phi$ is bounded with compact support and because $f$ is assumed to be integrable on compact subsets of $\Omega$

Comments

1. If $f$ is of class $\mathcal C^1$ then the gradient of $f$ is the weak derivative of $f$ in view of Gauss Green Theorem. So the weak derivative is an extension of the classical derivative.
2. The weak derivative is unique (as an element of the Lebesgue space $L^1_{\mathrm loc}$ in view of a result about locally integrable functions.
3. The same definition can be given for functions with complex values.