variation of laplacian on a compact riemannian manifold

Question

I have a problem with a part of the proof in the article "prescribing curvature, Kazdan-Warner" of Lemma 3.2. In this lemma is required to compute the variation of the Laplacian operator, the authors say that $$\Delta'u=\frac{d}{dt}\Delta_{g_t}u|_{t=0}=-h^{ij}u_{,ij}+\bigg{(}\frac{1}{2}h^{i,j}_{i}-h^{ij}_{,i}\bigg{)}u_{,j}$$ $$k'=\frac{d}{dt}k_{g_t}|_{t=0}=-\Delta h^i_i+h^{ij}_{,ij}-h^{ij}Ric(g)^i_j$$ where $g_t=g-th$. Did someone do this calculations? I don't understand very well the tensor calculus, if it is possible, could you write it without this notation? Thanks.