I've been reading the book Gauge, Fields, Knots and Gravity by Baez.
A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties:
$V(f+g)=V(f) + V(g)$.
$V(\alpha f)= \alpha V(F)$.
$V(fg) = V(f)g(p) + V(g)f(p)$.
Can someone explain me what is the physical interpretation of tangent vectors and the above definition?