The space of derivations on
Let $A$ be an algebra over a field $A$ k$, such as $C^{\infty}(M)$ for $k = \mathbb{R}$. You should be thought think of "points" as the Lie algebra meaning $k$-algebra homomorphisms $A \to its "Lie group" of automorphisms, k$ and the Leibniz rule is the infinitesimal version of the fact that an automorphism "one-parameter families of an algebra preserves products. In geometric situations points" as meaning $A$ is the space of smooth k$-algebra homomorphisms $A \to k[t]$ (or at least in a more algebraic , or etc.) functions on some space and the infinitesimal version of an automorphism setting). The intuitive meaning of a space is a "tangent vectorfield. See " is "infinitesimal one-parameter family of points," and algebraically this blog post for means a thorough discussionmorphism $A \to k[t]/t^2$. (There The Leibniz rule is also a local story about tangent vectors but it follows along equivalent to the same lines.)statement that this homomorphism preserves products.
The more fundamental property is really the chain rule, but note that linearity and the Leibniz rule are equivalent to the chain rule for polynomials, and in an algebraic setting polynomials are the only things available. In a less algebraic setting, e.g. smooth manifolds, it's actually more natural to require the chain rule for all smooth functions; this is closely related to the idea that $C^{\infty}(M)$ is not really an algebra but a smooth algebra.
