Let $M$ be an $n-$dimensional smooth manifold. Usually, a tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties(linear derivation):
$V(f+g)=V(f) + V(g)$.
$V(\alpha f)= \alpha V(F)$.
$V(fg) = V(f)g(p) + V(g)f(p)$.
However, in Hirsch's differential topology, a tangent vector to $M$ is an equivalence class $[x,i,a]$ of triples $(x,i,a)\in M\times \Lambda \times \mathbb R^n$ where $\lambda$ is the index set for charts $(U_i,\phi_i)$, under the equivalence relation: $$[x,i,a]=[y,j,b]$$ if and only if $x=y$ and $$D(\phi_j \phi_i^{-1})(\phi_i(x))a=b.$$
How to prove these two definitions are equivalent, if they indeed are?