Here is a scalar field and is a vector field.
Rewrite your expression in component form:
where is the fully antisymmetric symbol which is 0 if any indices are the same, 1 if ijk is cyclic to xyz and -1 if ijk is cyclic to xzy.
Concentrate on what the derivative is doing. This is just like d(uv) = (du)v + u(dv):
So the proof goes: