Why is $\frac{1}{2}g^{im}\frac{\partial g_{im}}{\partial x^k}=\frac{1}{2g}\frac{\partial g}{\partial x^k}$ true

Question

The following Wiki page has the following formula: $$\frac{1}{2}g^{im}\frac{\partial g_{im}}{\partial x^k}=\frac{1}{2g}\frac{\partial g}{\partial x^k}$$ I don't understand how this is. I am interpreting $g$ as the determinant of the metric $g$.

Answer 1

The variation of the determinant of a symmetric matrix is $$\delta \det M =\det M \,\mbox{Tr}(M^{-1}\delta M)$$ Using this formula for the metric tensor as well as $\delta^i_j = g^{ik}g_{kj}$, we get $$\frac{1}{g}\delta g=g^{\mu \nu}\delta g_{\mu \nu}$$