I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that $\partial_{\mu} g^{\nu\rho} = -g^{\nu\kappa}g^{\rho\lambda}\partial_{\mu}g_{\lambda\kappa}$
LHS seems to me like it should be zero, but other than that I have no clue.
available info is that all Greek letters are from 0 to 3.
Also, do you know any resources with solved problems like that I can work on?