I need some clarification with the problem in this post (unfortunately I cannot comment there, so I'm asking it as a separate question).
The problem in short:
We have an infinitesimal coordinate change $ x'^{\mu} \rightarrow x^{\mu}+\alpha k^{\mu} $.
Then, the metric will also change (taken from the answer of @Prahar): $$ g_{\mu \nu}^{\prime}\left(x^{\prime}\right)=\frac{\partial x^{\rho}}{\partial x^{\prime \mu}} \frac{\partial x^{\sigma}}{\partial x^{\prime \nu}} g_{\rho \sigma}(x). $$
The goal is to derive the following expression for the new metric:
$$ g_{\mu \nu}^{\prime}(x)=g_{\mu \nu}(x)-\alpha\left(g_{\mu \rho} \partial_{\nu} k^{\rho}+g_{\nu \rho} \partial_{\mu} k^{\rho}+k^{\rho} \partial_{\rho} g_{\mu \nu}\right)+\mathcal{O}\left(\alpha^{2}\right). $$
The part where I'm stuck: how exactly do we substitute $x'^\mu$? I started out with $x^\mu = x'^\mu - \alpha k^\mu $, which means $$\frac{\partial x^{\rho}}{\partial x^{\prime \mu}} = \delta^\rho_\mu - \frac{\partial \epsilon k^{\alpha}}{\partial x^{\prime \mu}}$$ but I have no idea what to do after this step.