Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule?

Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\infty}(M)$. Does it necessarily admit an elliptic operator $D$ which satisfies $D(\{f,g\})=\{D(f),g\}+\{f,D(g)\}$?