Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 68 / 3 / a

Differentiate $g^{ac}{g}_{cb}={\delta }_b^a$ while holding the coordinate system fixed. The variation of inverse metric obeysMultiplication by the inverse metric tensor givesThis is the matrix inverse derivative written with tensor indices. A variation of the metric tensor is symmetric, so the right-hand side is symmetric in $c,d$.