Let $f: X \rightarrow S$ be a morphism of schemes which is locally of finite type. We say $f$ is unramified if for every $x \in X$, we have $(\Omega_{X/S})_x = 0$, where $\Omega_{X/S}$ is the sheaf of Kahler differentials of $f$.
Let $L/K$ be a finite extension of $p$-adic fields, with $B = \mathcal O_L, A = \mathcal O_K$. Let $X = \operatorname{Spec} B, S = \operatorname{Spec} A$. I want to understand why $f: X \rightarrow S$ is an unramified morphism of schemes if and only if $L/K$ is unramified in the usual sense, i.e. the maximal ideal of $A$ generates the maximal ideal of $B$.
When $x$ is the generic point of $X$, we have $(\Omega_{X/S})_x = \Omega_{B_{(0)}/A_{(0)}} = \Omega_{L/K}$, which is zero since $L/K$ is finite separable.
When $x = \mathfrak P$ is the closed point of $X$, lying over the closed point $s = \mathfrak p$ of $S$, $(\Omega_{X/S})_x$ is just $\Omega_{B/A}$.
We have $\Omega_{B/A} = 0$ if and only if every $A$-derivation of $B$ into every $B$-module is trivial. So I want to show that $\mathfrak p$ generates $\mathfrak P$ if and only if every $A$-derivation of $B$ into every $B$-module is trivial.
Let $M$ be a $B$-module. Supposing that $\mathfrak p$ generates $\mathfrak P$, I want to show that if $\delta: B \rightarrow M$ is an $A$-derivation, then $\delta = 0$. Letting $\varpi \in A$ be a uniformizer of $B$, every element of $B$ can be written as $u \varpi^n$ for $n \geq 0$ and $u \in B^{\ast}$. Since $\delta(a) = 0$ for all $a \in A$, we have $\delta(\varpi^n) = 0$ by induction, so this direction is done if I can show that $\delta(u) = 0$ when $u \in B$ is a unit. So far I haven't been able to show this.
For the other direction, I would need to show that if $\mathfrak p$ does not generate $\mathfrak P$, then there is some $B$-module and some nontrivial $A$-derivation into it. I thought about doing something with the $B$-module $\mathfrak P/ \mathfrak P^e$, where $e \geq 2$ is the ramification index.
Is this a viable approach to prove what I want? Are there other ways to connect these two notions of being unramified?