Let $M$ be a manifold and $d$ the exterior derivative on forms. We extend $d$ to vector fields as follows: if $X = \sum_i X_i \partial_i$ in local coordinates, then
$$dX = \sum_i (dX_i) \partial_i$$
i.e. if $Y$ is another vector field then $dX(Y)$ is the vector field $\sum_i (dX_i)(Y) \partial_i$. I am trying to show that this is well defined, independent of coordinate changes. When I calculate it out explicitly I get a rather nasty expression which does not seem to simplify. A coordinate-free definition of $d$ would be ideal, but I would also be satisfied with a coordinate based calculation.