Covariant derivative along a curve vs pullback connection

Question

Let $\nabla$ denote the Levi Civita connection on a Riemannian manifold $M$. Suppose that $c$ is a curve and $X$ a smooth vector field. My question is: how are the following covariant derivatives $$ \bigl(\nabla_{\dot c(t)}X\bigr)(c(t)), $$ and $$ \bigl(\nabla_{\frac{\partial}{\partial t}}X\circ c\bigl)(t) $$ related? In the second case we consider the pullback connection (or material derivative). I am confused.