Derivations are determined by their values on linear functions

How are derivations of the $\mathbb R$ algebra of germs of differentiable real functions on a manifold completely determined by their values in germs of linear functions? Are derivations of more general algebras of functions determined similarly? Does this result have an anologue for derivations of abstract algebras?

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Hint: Take $f\in C^\infty_x$ and write $f=f(a)+f'(a)x+(f-f(a)-f'(a)x)$. Use the Leibniz rule to show that $f-f(a)-f'(a)x$ vanishes under any derivation.

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