Natural derivative of Vector Fields on manifolds

Question

I'm learning about connections and my book says that there is no natural derivative for a vector field on a manifold. Wouldn't it be possible to cook up a connection by just letting $\nabla_{v_p}X = \sum v_pa^i \frac{\partial}{\partial x^i}|_p$ where $\sum a^i \frac{\partial}{\partial x^i}|_p$ is the coordinate representation of $X$ about $p$ or do we run into the problem that this is dependent on the choice of coordinates? Apparently we are able to do this in $R^n$ but this is not the case for a general manifold? Thank you.