Let \(M\) be a smooth manifold, let \(\nabla\) be the connection in \(TM\) and \(\gamma\colon I\to M\) a geodesic with respect to \(\nabla\).
In the following we abbreviate
\begin{equation*} \nabla^n_{X,\ldots ,X}F = \nabla^n_XF. \end{equation*}Then for any smooth tensor field \(F\) on \(M\) and \(n\in \mathbb{N}\) we have
\begin{equation*} \nabla_{\gamma'}(\nabla^{n-1}_{\gamma'} F) = \nabla^n_{\gamma'}F. \end{equation*}Note, that compared to the second covariant derivative no additional term appears.