# Derivative of contravariant metric tensor with respect to covariant metric tensor

1. Nov 17, 2011

### PhyPsy

1. The problem statement, all variables and given/known data
Show that $$\frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d})$$

2. Relevant equations

3. The attempt at a solution
It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming up with: $$\frac{\partial[\delta^c{_a}\delta^d{_b}(g_a{_b})^-{^1}]}{{\partial}g_a{_b}}$$$$-\delta^c{_a}\delta^d{_b}(g_a{_b})^-{^2}=-(g^c{^d})^2$$
Maybe I am not applying the Kronecker deltas correctly.

Last edited: Nov 17, 2011