I'm trying to determine the equation of divergence for the given metric
$$ g_{ij} = \begin{bmatrix} u^2+v^2 & 0 & 0 \\ 0 & u^2+v^2 & 0 \\ 0 & 0 & u^2v^2 \end{bmatrix} $$
Which is the metric for a paraboloidal space.
The divergence of some vector, given in my textbook, is
$$ \nabla_{i}V^{i} = \partial_{i}V^{i} + \Gamma^{i}_{i j}V^{i} = \frac{1}{\sqrt{|g|}} \partial_i(\sqrt{|g|}\ V^{i} ) $$
Where $g = \det{g_{ij}}$
If I where to work it out, how would some parts in the derivative cancel out with the inverse determinant? I tried this formula to determine the divergence in spherical coordinates, but I also run into the same problem to how I cancel coefficients.