Let \(M\) be a smooth manifold, \(\nabla\) a connection in \(TM\) and \(\gamma\colon I\to M\) a maximal geodesic with initial point \(p\in M\) and \(\gamma'(0)\in T_pM\). Then the \(n\)-th derivative of a smooth real-valued function \(f\) along \(\gamma\) at \(0\in I\) is given by
\begin{equation*} (f\circ \gamma)^{(n)}=\nabla^n_{\gamma'}f. \end{equation*}Note, we use the convention \(\nabla^n_{\gamma'}f = \nabla^n_{\gamma',\ldots ,\gamma'} f\).