I am trying to derive the following equivalence:
$$ g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j}\frac{\partial f}{\partial x^k}+ \frac{1}{2}g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k} = g^{jk}\frac{\partial ^2 f}{\partial x^j \partial x^k}- g^{jk}\Gamma^l_{jk}\frac{\partial f}{\partial x^l} $$
I was thinking of using this equation for the Christoffel symbols:
$$ \Gamma^m_{ij}=\frac{1}{2}g^{mk}(\frac{\partial g_{ki}}{\partial x^j}+\frac{\partial g_{kj}}{\partial x^i}-\frac{\partial g_{ij}}{\partial x^k}) $$
EDIT after the suggestion in the comments
Now, when substituting this equation in the right hand side of the first equation, I get for the relevant term:
$$ g^{jk}\Gamma^l_{jk}\frac{\partial f}{\partial x^l}= g^{jk}( \frac{1}{2}g^{il} (\frac{\partial g_{jl}}{\partial x^k} +\frac{\partial g_{ik}}{\partial x^j} -\frac{\partial g_{jk}}{\partial x^i}) \frac{\partial f}{\partial x^l} ) $$ exhanging $i$ with $j$ and $l$ with $k$ and reordering terms:
$$ = g^{jk}( \frac{1}{2}g^{il} (\frac{\partial g_{ik}}{\partial x^l} +\frac{\partial g_{jl}}{\partial x^i} -\frac{\partial g_{il}}{\partial x^j}) \frac{\partial f}{\partial x^k} ) $$
$$ = g^{jk}( \frac{1}{2}g^{il} (\frac{\partial g_{ik}}{\partial x^l} +\frac{\partial g_{jl}}{\partial x^i}) \frac{\partial f}{\partial x^k} ) - \frac{1}{2} g^{jk} g^{il} \frac{\partial g_{il}}{\partial x^j} \frac{\partial f}{\partial x^k} $$
$$ = \frac{1}{2}g^{jk}g^{il} \frac{\partial g_{ik}}{\partial x^l} \frac{\partial f}{\partial x^k} + \frac{1}{2}g^{jk}g^{il} \frac{\partial g_{jl}}{\partial x^i} \frac{\partial f}{\partial x^k} - \frac{1}{2} g^{jk} g^{il} \frac{\partial g_{il}}{\partial x^j} \frac{\partial f}{\partial x^k} $$ Now, using the suggestion that
$$g^{ij}g_{jk} = \delta^i_k$$ gives $$\frac{\partial (g^{ij}g_{jk})}{\partial l}=0$$ and so $$g^{ij}\frac{\partial g_{jk}}{\partial l}=-g^{jk}\frac{\partial g^{ij}}{\partial l}$$
the previous formula becomes
$$ - \frac{1}{2}g^{jk}g_{ik} \frac{\partial g^{il}}{\partial x^l} \frac{\partial f}{\partial x^k} - \frac{1}{2}g^{jk}g_{jl} \frac{\partial g^{il}}{\partial x^i} \frac{\partial f}{\partial x^k} - \frac{1}{2} g^{jk} g^{il} \frac{\partial g_{il}}{\partial x^j} \frac{\partial f}{\partial x^k} $$
$$ - \frac{1}{2} \frac{\partial g^{il}}{\partial x^l} \frac{\partial f}{\partial x^k} - \frac{1}{2} \frac{\partial g^{il}}{\partial x^i} \frac{\partial f}{\partial x^k} - \frac{1}{2} g^{jk} g^{il} \frac{\partial g_{il}}{\partial x^j} \frac{\partial f}{\partial x^k} $$
$-\frac{\partial g^{jk}}{\partial x^j}\frac{\partial f}{\partial x^k}- \frac{1}{2}g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k}$
where I use the fact that $g_{ik}=g_{ki}$ and $g^{ik}=g^{ki}$ and that I can rename indexes on a term that fully "contracts" or if they are not used elsewhere.
Do my assumptions make sense?
thanks