Let \(M\) be a smooth manifold and \(\nabla\) a connection in \(C^\infty(M)\). The derivative of a smooth real-valued function \(f\) along a smooth curve \(\gamma\colon I\to M\) at some point \(t_0\in I\) may be expressed in terms of the connection
\begin{equation*} (f\circ \gamma)'=\nabla_{\gamma'} f, \end{equation*}where \(\gamma'\) denotes the velocity of \(\gamma\).