Let $M$ be a differential manifold and $\alpha \in {\bf R}$. I define an $\alpha$-derivation $X_{\alpha}$ as an application of the smooth functions such that:

$$X_{\alpha} (fg)=X_{\alpha}(f)g^{\alpha}+f^{\alpha}X_{\alpha}(g)$$

$$X_{\alpha}(f+g)^{1/\alpha}=X_{\alpha}(f)^{1/\alpha}+X_{\alpha}(g)^{1/\alpha}$$

The definition makes sense because:

$$X_{\alpha}((fg)h)=X_{\alpha}(f(gh))=X_{\alpha}(fgh)=$$

$$=X_{\alpha}(f)(gh)^{\alpha}+f^{\alpha}X_{\alpha}(g)h^{\alpha}+(fg)^{\alpha}X_{\alpha}(h)$$

What are the $\alpha$-derivations of the real smooth functions?