I'm looking for an explanation for the meaning of the terms on the right side of the following:
\begin{eqnarray}\frac{d}{dt} \Big( \bar{R}_{ij} \Big)\Big|_{t=0} = \dot{\Gamma}^k_{ij;k} - \dot{\Gamma}^{k}_{ki;j},\end{eqnarray} where $$\dot{\Gamma}^k_{ij} = \frac{d}{dt}\Big( \bar{\Gamma}^{k}_{ij} \Big)\Big|_{t=0} = \frac{1}{2}g^{kl}(h_{il;j} + h_{jl;i} - h_{ij;l})$$ and the bar denotes the dependence on $\bar{g} = g + th$, for a metric $g$, a symmetric $(0,2)$-tensor $h$, and $t \in (\varepsilon,\varepsilon)$ (so we have a one-parameter family of metrics on a fixed manifold $M$).
EDIT: $\dot{\Gamma}$ is a tensor. Thus it makes sense to take the covariant derivative. That is,
\begin{align*} \dot{\Gamma}^k_{ij;\ell} = (D_{\partial_{\ell}}\dot{\Gamma})(dx^k,\partial_i,\partial_j) &= \dot{\Gamma}^k_{ij,\ell} - \dot{\Gamma}(D_{\partial_{\ell}}dx^k, \partial_i, \partial_j) - \dot{\Gamma}(dx^k,D_{\partial_{\ell}}\partial_i, \partial_j) - \dot{\Gamma}(dx^k,\partial_i, D_{\partial_{\ell}}\partial_j)\\ &= \dot{\Gamma}^k_{ij,l} -\dot{\Gamma}([\partial_{\ell}(dx^k(\partial_n)) - dx^k(D_{\partial_{\ell}}\partial_n)]dx^n, \partial_i,\partial_j) -\Gamma^n_{\ell i}\dot{\Gamma}^k_{nj} - \Gamma^n_{\ell j}\dot{\Gamma}^k_{in}\\ &= \dot{\Gamma}^k_{ij,l} +\Gamma^k_{\ell n}\dot{\Gamma}^n_{ij} -\Gamma^n_{\ell i}\dot{\Gamma}^k_{nj} - \Gamma^n_{\ell j}\dot{\Gamma}^k_{in}. \end{align*}