I know that on a Lorentzian manifold the Laplace-Beltrami operator of a function $\phi\in C^\infty(M)$ is defined as $$ \triangle_g \phi = \text{div}(\text{grad}\phi). $$ Now I've come across the following expression in local coordiantes $$ \triangle_g \phi = g^{\alpha\beta}(\partial_\alpha\partial_\beta\phi+\Gamma^\lambda_{\alpha\beta}\partial_\lambda\phi) $$ but I was not able to derive it (nor to find a derivation online). Why is this expression true? How to derive it from the definition?
(I am also not sure if the signs are correct the way I wrote it.)