I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ is the inverse matrix to $g_{ij}$ and $g$ denotes the determinant.
$$ \Gamma_{\,jl}^j = \frac{\partial}{\partial x^l} \log \sqrt{\vert g \vert} $$
Here is what I have so far: $$ \Gamma^j_{\;jl} = \frac{1}{2} g^{jk} \left( \frac{\partial }{ \partial x^l} g_{kj} + \frac{\partial}{ \partial x^j} g_{kl} - \frac{\partial}{ \partial x^k} g_{jl} \right) = \frac{1}{2} g^{jk} \frac{\partial }{ \partial x^l} g_{kj} $$
The second and the third term cancel each other out. Now as stated here (it's about the sixth equation, after the line starting with "The contracting relations..."
$$ \frac{1}{2} g^{jk} \frac{\partial }{ \partial x^l} g_{kj} = \frac{1}{2g} \frac{\partial}{ \partial x^l} g = \frac{\partial}{ \partial x^l} \log \sqrt{\vert g \vert} $$
I don't get the second last step. I suspect it has got something to do with the Laplace formula and the Cramer Rule. However i seem not to be able to figure it out, the derivation is bugging me.
Any hints are appreciated!