TITLE: Why is the Leibniz rule a definition for derivations?
QUESTION [26 upvotes]: In differential geometry, the tangent space is defined as a generalization of directional derivatives, which in turn are defined as functionals following Leibniz's product rule.
I understand all the proofs, but what is the intuition behind choosing the product rule to capture the notion of derivatives, rather than any other property?

REPLY [7 votes]: The derivation property
is the simplest possible property; all other are elaborations:
Let $M$ be a smooth manifold. Then we have in the category of commutative $\mathbb R$-algebras with unit (see p 296  of the reference below
for more details):

$Hom(C^\infty(M,\mathbb R),\mathbb R) = M$
$Hom(C^\infty(M,\mathbb R),\mathbb D) = TM$ where $\mathbb D$ (generated by $1,\epsilon$ with $\epsilon^2=0$) is the algebra of study numbers.
$Hom(C^\infty(M,\mathbb R),\mathbb D\otimes \mathbb D) = T(TM)$
For any finite dimensional commutative $\mathbb R$-algebra $A$ we have:
$Hom(C^\infty(M,\mathbb R),A) = Hom(C^\infty(M,\mathbb R),W(A))$ is a smooth manifold and describes a product preserving covariant endofunctor $T_{W(A)}$ on the category of smooth manifolds and smooth mappings. Here $W(A)$ is the subsalgebra of $A$ generated by all idempotent and nilpotent elements.  Up to some covering space phenomena all product preserving functors are of this form. $W(A)$ is called the Weil algebra of $A$. 
Then $T_A\circ T_B = T_{A\otimes B}$; thus composition of product preserving functors is naturally commutative.
Now consider only algebras $A$ with $A=W(A)$, and with exactly one idempotent 1 (for simplicity's sake). Then $A=\mathbb R.1\oplus N$ for a nilpotent ideal $N$.
Moreover, $T_A(M)\to M$ is a fiber bundle. 
Elements $\xi_x$ in the fiber over $x\in M$ correspond exactly to $\mathbb R$-linear mappings
$C^\infty(M,\mathbb R) \to N$ satisfying 
$\xi_x(f.g) = \xi_x(f).g(x) + f(x).\xi_x(g) + \xi_x(f).\xi_x(g)$.
Smooth section $\xi$ of this bundle correspond exactly to 
$\mathbb R$-linear mappings $C^\infty(M,\mathbb R) \to C^\infty(M,\mathbb R)\otimes N$ such that $\xi(f.g) = \xi(f).g + f.\xi(g) + \xi(f).\xi(g)$ (expansion property).
The Leibnitz rule is the simples possible expansion property.
Addition of vector fields is replaced by a non-commutative nilpotent group multiplication. 

Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993), (pdf)