Suppose that $A$ is an $R$-algebra, with both $R, A$ integral domains. Let $B = S^{-1}A$ be a localization of $A$.
Let $$\partial_A: A \to M$$ $$\partial_B: B \to S^{-1}M$$ be an $R$-derivation on $A$ and its unique extension to $B$.
In this question it is asked whether, in the case of $A=M$, the obvious inclusion $$\mathrm{ker}(\partial_A) \subset \mathrm{ker}(\partial_B)$$ is an equality.
A counterexample was provided: the derivation $D = x\partial_x + y\partial_y$ on $\mathbb{C}[x,y]$.
My question regards the $\textit{universal derivation}$ $$d:A \to \Omega_{A/R}$$
and when its constants are preserved under localization. In a sense this captures the common constants under all derivations.
If $R$ is not a field, and $B$ contains some inverses of $R$ (measured by $S_0 := S \cap R$), then this will clearly enlarge the constants. Further, if $S_1 := S \cap \mathrm{ker}(d_A)$ $$\mathrm{ker}(d_A) \subset S_0^{-1}\mathrm{ker}(d_A) \subset S_1^{-1}\mathrm{ker}(d_A) \subset \mathrm{ker}(d_B)$$ So I would like to ask, under what conditions will this second inclusion be an equality: $$S_1^{-1}\mathrm{ker}(d_A) = \mathrm{ker}(d_B)$$
which I might paraphrase as: when are the constants of the localization no larger than expected from localizing?
I am interested in the general answer (and even how much can be said in positive characteristic), but the most important case for me is when $\mathrm{Frac}(A)/\mathrm{Frac}{R}$ is a field extension of transcendence degree $1$.
The motivation comes from comparing relative de Rham cohomology of a scheme with the cohomology of open subschemes.