Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

First the slick method: write $\bar g_{ik}$ for the value of $g_{ik}$ at the specific $x^l = \bar x^l$ (so $\bar g_{ik}$ is constant). Let $x^l$ be a function of $\epsilon$ that is a perturbation of $\bar x^l$, so that $x^l = \bar x^l$ at $\epsilon=0$.

Let $$h_{ij} = \bar g^{ik} g_{kj} . \tag1$$ Then $h_{ij}$ is a perturbation of the identity: $$h_{ij} = \delta_{ij} + \epsilon \frac{\partial h_{ij}}{\partial \epsilon} + o(\epsilon).$$ Then it is easily computed that $$ \frac{\partial h}{\partial \epsilon} = \frac{\partial h_{ii}}{\partial \epsilon} + o(\epsilon)$$ Differentiate $(1)$ with respect to $\epsilon$ $$ \frac{\partial h_{ii}}{\partial\epsilon} = \bar g^{ik} \frac{\partial g_{ki}}{\partial \epsilon} $$ Also $h = \bar g^{-1} g$. Hence $$ \bar g^{-1} \frac{\partial g}{\partial \epsilon} = \bar g^{ik} \frac{\partial g_{ki}}{\partial \epsilon} + o(\epsilon). $$ In particular, at $\epsilon = 0$ we can remove the overlines, and we get: $$ g^{-1} \frac{\partial g}{\partial \epsilon} = g^{ik} \frac{\partial g_{ki}}{\partial \epsilon}. $$

Next the direct way: so $$g = \sum_\pi\sigma(\pi) \prod_i g_{i\pi(i)}.$$ Here the sum is over permutations $\pi$ of $\{1,\dots,n\}$. Then $$\frac{\partial g}{\partial x^l} = \sum_\pi \sigma(\pi) \sum_i \frac{\partial g_{i\pi(i)}}{\partial x^l} \prod_{j \ne i} g_{j\pi(j)} \\ =\sum_{ik} \frac{\partial g_{ik}}{\partial x^l}\sum_{\pi:\pi(i)=k} \prod_{j\ne i}\sigma(\pi) g_{j\pi(j)} \\ =\sum_{ik} \frac{\partial g_{ik}}{\partial x^l} g \ g^{ki}. $$