To expand on Mariano's comment, let $A$ be, say, an $\mathbb{R}$-algebra and $\phi : \mathbb{R} \to \text{Aut}(A)$ a one-parameter group of automorphisms of it. Suppose that we can make sense of the derivative of $\phi$. Then $d \phi_0 : \mathbb{R} \to \text{End}(A)$ is linear, and differentiating the condition that $\phi$ is a family of automorphisms gives that $d \phi_0(1)$ satisfies the Leibniz identity. In other words, the Leibniz identity is the infinitesimal analogue of preserving multiplication. 

In fact $A$ need not be associative for this argument to go through. This makes the above a nice way to think about, for example, the Jacobi identity for a Lie bracket: it says precisely that $[x, -]$ is a derivation, and this is simply because it is the infinitesimal version of the adjoint action.