I have come across the following two different definitions for "derivation at a point" (for both definitions, assume M is a smooth manifold and $p \in M$):
Def 1: A derivation at p is an $\mathbb{R}$-linear map $\delta : C^{\infty}(M) \rightarrow \mathbb{R}$ satisfying the Leibniz identity $\delta(fg) = \delta(f) g(p) + f(p) \delta(g)$
Def 2: A derivation at p is an $\mathbb{R}$-linear map $\delta : C_p^{\infty}(M) \rightarrow \mathbb{R}$ satisfying the Leibniz identity $\delta(fg) = \delta(f) g(p) + f(p) \delta(g)$
Both are claimed to define the same notion of the tangent space $T_pM$, even though one is defined as maps from the algebra of functions and the other from the algebra of germs of functions.
Question: How can I construct an isomorphism between the vector space of derivations according to definition 1 and the vector space of derivations according to definition 2?
My idea has been to construct a map which takes functions to their germs as a quotient map; however, I can't figure out how to show that germ(g) = germ(f) $\implies \delta(g) = \delta(f)$ for all definition 1 derivations. Any help would be much appreciated.