How to check whether a linear map on integral domains is a formal derivative

Question

I have an elementary question on formal derivatives. Assume $A=K[X,Y,Z]/I$ is an integral domain (for example $I$ is a prime ideal and K is the field of rationals). Let $d:A\to A$ be a linear map. Is there a standard test to check whether $d$ is a formal derivative (i.e. satisfies Leibniz rule $d(a\cdot b)=d(a)\cdot b+d(b)\cdot a$)? I imagine one has to check that $d(p(X,Y,Z))=0$ for $p(X,Y,Z)$ a polynomial in I. Is this necessary and sufficient? Are there other tests?