Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$.

Let $I$ be the ideal defining the singular locus in Spec $R$. (With the reduced subscheme structure, or defined using minors of a Jacobian matrix, again whatever helps.)

Is it obvious and/or true that any derivation $d:R \to R$, i.e. additive map satisfying the Leibniz rule $d(ab) = a\ db + b\ da$, has $dI \leq I$?

Morally, $d$ is defining an infinitesimal automorphism of Spec $R$, and the singular locus should be preserved by automorphisms. So I would have hoped that there was a mindless proof using the Jacobian, but I haven't found one. As usual, a reference would be even better than a proof.