I am asked to prove that the differential equations of geodesics on an open set of a pseudoRiemannian manifold $(M, g)$ of dimension $n$ where $g_{ij}=0$ if $i \neq j$ are given by
$$ \frac{d}{ds}(g_{kk}\frac{dx^{k}}{ds})=\frac{1}{2} \sum_{i} \frac{\partial g_{ii}}{\partial x^{k}}(\frac{dx^{i}}{ds})^2, 1\leq k \leq n $$
I know that the general formula is given by
$$ \frac{d^{2}x^{k}}{ds^{2}}+ \sum_{i} \sum_{j} \Gamma^{k}_{ij} \frac{dx^{i}}{ds} \frac{dx^{j}}{ds})=0, 1\leq k \leq n $$
And I have computed that the non-zero Chrystoffel symbols are:
$$ \Gamma^{k}_{ik} = \frac{1}{2}( \frac{\partial g_{kk}}{\partial x^{i}}) \frac{1}{g_{kk}}$$ $$ \Gamma^{k}_{kj} = \frac{1}{2}( \frac{\partial g_{kk}}{\partial x^{j}}) \frac{1}{g_{kk}} $$ $$ \Gamma^{k}_{jj} = \frac{1}{2}( -\frac{\partial g_{jj}}{\partial x^{k}}) \frac{1}{g_{kk}} $$ or
$$ \Gamma^{k}_{jj} = \frac{1}{2}(\frac{\partial g_{jj}}{\partial x^{j}}) \frac{1}{g_{kk}} $$
However, I can't seem to reach the expression I'm asked for from that.