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 \(\gamma\).
Proof
Let \((x^i)\) be local coordinates. We write the component functions of \(\gamma\) as \(\gamma(t)=(x^1(t),\ldots ,x^n(t))\). Then using the generalization of the chain rule for multi-variable functions, we obtain
\begin{equation*} (f\circ \gamma)'=\partial_i f \, \dot{x}^i= df(\gamma')=\nabla_{\gamma'} f. \end{equation*}