  We prove that given a locally integrable function $f$ on an open set of an
Euclidean space the distributional derivative $Xf$ with respect to a locally
Lipshitzian vector field $X$ is locally integrable if, and only if, the
function $f$ admits a locally integrable upper gradient along the vector field
$X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is
the least upper gradient of the function $f$. Applications to systems of
locally Lipshitzian vector fields are given.
