Does the pushforward $T_a(X)$ of a vector field $X$ on $M$ map to $T_a(X)$ again?

Question

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary and $X:M\to\mathbb R^d$ with $$X(a)\in T_a\:M\;\;\;\text{for all }a\in M\tag1.$$

Let $a\in M$. By definition, the pushforward $T_a(X)$ is a map from $T_a\:M$ to $T_{X(a)}\mathbb R^d=\mathbb R^d$, but can we somehow show that it actually maps to $T_a\:M$?