How to differentiate by a vector ?

Is it possible to differentiate a scalar( in this case a Lagrangian) by a vector ?
If the Lagrangian is r.A ie. the scalar product of vectors r and A what is ∂L/∂r ? My notes say it is A but how ?
My notes also say that ∂L/∂r is the same as ∇L. Is this correct ? or is it sloppy notation ? or is it because the vector is treated as a generalised coordinate which acts as a scalar ?
Thanks