Nigel Hitchen, in his notes on differential manifolds gives a definition of the derivative of a smooth map between two manifolds that appears to assume the the following assertion is self-evident:
Given two manifolds $M$ and $N$, a smooth map $F: M \rightarrow N$, and a tangent vector $X_a$ at the point $a\in M$, the map $X'_{F(a)}: C^\infty(N) \rightarrow C^\infty(\mathbb{R})$ defined by $$X'_{F(a)}(f)=X_a(f\circ F)$$ for $f\in C^{\infty}(N)$ is a tangent vector at $F(a)$.
I see that $X'(1) = 0$ but how about the Leibnitz rule?
If $\{f,g\} \subset C^\infty(N)$ then $\{f\circ F,g\circ F, fg\circ F\} \subset C^{\infty}(M)$ and $fg\circ F = (f\circ F)(g\circ F)$, so $X'_{F(a)}$ inherits the Leibnitz rule from $X_a$.