It is well known that we can define the tangent space of a manifold $M$ at a point $p\in M$ as the set of speeds, at time $0$, of curves $\alpha : (-\varepsilon, \varepsilon) \to M$ such that $\alpha(0) = p$. What if $M$ is a manifold with boundary and $p$ is a boundary point? Is it correct to say that
$$T_p M = \{ \alpha'(0^+) : \alpha : (-\varepsilon, \varepsilon) \to M \text{ is smooth and } \alpha(0) = p \} \cup \{ \alpha'(0^-) : \alpha : (-\varepsilon, \varepsilon) \to M \text{ is smooth and } \alpha(0) = p \}$$
where $\alpha(0^+)$ and $\alpha(0^-)$ are directional derivatives?