Solution
= Solution
Write $h_{ab}=\delta g_{ab}$ and $h=g^{ab}h_{ab}$. Standard <metric variation> gives
$$
\boxed{\delta\sqrt{-g}=\frac12\sqrt{-g}\,h},
\qquad
\boxed{\delta g^{ab}=-h^{ab}},
$$
and variation of metric compatibility gives
$$
\boxed{\delta\Gamma^a{}_{bc}
=\frac12g^{ad}
(\nabla_bh_{cd}+\nabla_ch_{bd}-\nabla_dh_{bc})}.
$$