Consider then a the vector fields $\partial_{b}$ and $V$; $\partial_{b}$ are the (covariant) basis vectors and $V$ is a contravariant vector field. With a (affine) connection $\nabla$ we can therefore define two derivatives:
\begin{equation} \nabla_{\partial_{b}} V = (\partial_{b}V^{a} + \Xi^{a}\hspace{0.1mm}_{bc}V^{c}) \partial_{a} =\nabla_{b}V^{a} \partial_{a} \tag{1} \end{equation}
\begin{equation} \nabla V = (\partial_{b}V^{a} + \Xi^{a}\hspace{0.1mm}_{bc}V^{c}) \partial_{a} \otimes \mathrm{d}x^{b} =\nabla_{b}V^{a} \partial_{a} \otimes \mathrm{d}x^{b} \tag{2} \end{equation}
My question is: what is the difference between $\nabla_{\partial_{b}} V$ and $\nabla V$ ?