Characterization of q-derivations

Question

Let $M$ be a manifold and $q$ a diffeomorphism. If $X$ is a linear application of the functions such that:

$$X(fg)(x)=X(f)(x)g(q(x))+f(x)X(g)(x)$$

with $f,g$ two smooth functions, then have we:

$$X(f)(x)=\alpha (x)(f(q(x))-f(x))$$

with $\alpha$ a function?

Answer 1

If $g(q(x))=0$, we have $X(fg)(x)=f(x)X(g)(x)$, so that $X(g)(x)=a(x)g(x)$ ; and if $f(x)=0$, we have $X(fg)(x)=X(f)(x)g(q(x))$, so that $X(f)(x)=b(x)f(q(x))$. We deduce that:

$$X(f)(x)=\alpha (x)(f(q(x))-f(x))$$