How do I go about proving
$\partial_{\mu} g^{\nu \rho}=-g^{\nu \sigma}g^{\rho \lambda}\partial_{\mu} g_{\sigma \lambda}$?
I've tried using the covariant derivative and the Christoffel symbols but it seems to be to no end in finding this specific relation.
Any ideas?
(Just to make this question answered)
Observe that for any invertible matrix
$$ A^{-1} A = I $$
where $I$ is the identity matrix. Now, let $A(t)$ be a parametrised family of invertible matrices, you have by the product rule:
$$ (\partial_t A^{-1}) A + A^{-1} \partial_t A = 0 $$
since the identity matrix is independent of $t$. Therefore
$$ (\partial_t A^{-1}) A = - A^{-1} \partial_t A \implies \partial_t A^{-1} = - A^{-1} (\partial_t A ) A^{-1} $$
writing out the matrix multiplication in index notation you get exactly the expression you wanted.