Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about this I will edit the post). I'm using the notations of the book "Liu - Algebraic geometry and arithmetic curves".
Let $(X,\mathscr O)$ be a Noetherian and geometrically integral scheme over a generic field $k$. Moreover let $K$ be the field of functions of $X$, $p\in X$ any closed point, and $\mathscr O_p$ the local ring at $p$.
At page 14 of Serre's book "Algebraic groups and class fields" there is the following equation (here I'm translating the notation in a more comprensible way):
$$\Omega^1_{K|k}=\Omega^1_{\mathscr O_p|k}\,\otimes_{\mathscr O_p} K$$
Could you please explain where this equality comes from?
You can find this in Eisenbud, "Commutative Algebra with a view toward Algebraic Geometry". It is Proposition 16.9 which says that
$$\Omega_{S[U^{-1}]|R} = \Omega_{S|R} \otimes_S S[U^{-1}]$$
for rings $S/R$ and $U$ a multiplicatively closed subset of $S$. In your case $R=k$, $S=\mathcal{O}_p$ and $U = S- (0)$, so that $S[U^{-1}] = K$.
As Eisenbud phrases it: "Localization of upper argument commutes with formation of differentials".