I am reading a lecture on Riemannian geometry in which it is written that, for a differentiable manifold $M$ and a differentiable curve $v \, : \, I \, \longrightarrow \, M$ defined on an interval $I$, we have (by definition) :
$$D_{t}v \cdot \left( \frac{\partial}{\partial t} \right) = v'(t)$$
I do not really understand what $\frac{\partial}{\partial t}$ is.