Derivations of intersections with disjoint spectra

Let $A_1,A_2\subseteq\mathbb{K}[x]$ be finite-codimension subalgebras whose spectra have no common elements, and set

$$A=A_1\cap A_2.$$

Intersection-derivation conjecture. The set of derivations of $A$ is the union of the derivations of $A_1$ and $A_2$, restricted to $A$. This conjecture concerns how derivations behave when two subalgebras with disjoint spectra are intersected; the preceding special cases support the proposed description, while the general assertion is left open.