I am again having a hard time solving an "easy" exercise of the lecture note by Ed Segal on the theory of manifold. (http://www.homepages.ucl.ac.uk/~ucaheps/papers/Manifolds%202016.pdf)
The elaborated statement is as follows.
$\bullet$ A linear map $\partial : C^\infty(X)\rightarrow \mathbb{R}$ is a derivation at $x$ iff $\partial$ vanishes on the subspace $R_x(X)\subset C^\infty(X)$ of functions having rank zero at $x$.
The definition of a derivation at $x$ is a linear map $\partial : C^\infty(X) \rightarrow \mathbb{R}$ obeying a Leibniz product rule.
Could somebody please prove the "if" part of the above proposition?
Thank you so much in advance.