Inverse metric volume variation ($\delta \sqrt{|g|}=-\frac{1}{2}\sqrt{|g|}{g}_{ab}\delta g^{ab}$)

For a nondegenerate metric tensor, differentiating $g_{ac}{g}^{cb}={\delta }_a{}^b$ gives $\delta g_{ab}=-g_{ac}{g}_{bd}\delta g^{cd}$. Combining this with the Jacobi determinant derivative formula proves the displayed volume-density variation. It is the determinant contribution when defining a stress-energy tensor by varying the inverse metric tensor.