Let $(M,\nabla)$ be a Riemannian manifold with the Levi-Cevita connection. Let $c:I \rightarrow M$ be a differentiable curve in $M$, and let $\frac{D}{dt}$ be the covariant derivative. Then
$\frac{D}{dt}(\frac{dc}{dt})=\nabla_{c'(t)}c'(t)$
I am confused about how we get this equality. I know it follows somehow from the following fact:
If $V$ is a vector field along $c$ that is induced by a vector field $Y \in \mathfrak{X}(M)$, i.e. $V(t)=Y(c(t))$ then $\frac{DV}{dt}=\nabla_{c'(t)}Y$.
This arises in do Carmo's textbook and he says it follows by "extending $c'(t)$ to a neighborhood of $c(t)$ in $M$". But this is confusing to me because on page 43 do Carmo says that $c$ cannot necessarily be extended to a vector field on an open set of $M$.
I know from the Smooth Extension Lemma that we can necessarily extend a vector field along only closed subsets to any open subset of the manifold.
So, I would appreciate it if someone could explain clearly how this works.