Willy Liu answered this in a comment. By EGA 4v4 Corollary 17.11.4, a necessary and sufficient condition that $X \to Y$ be smooth at $x \in X$ is that there be an open neighborhood $U \subset X$ of $x$, and an étale morphism $g: U \to \mathbb{A}^n_Y$ for some $n$. Étale maps are unramified, so the map $T_x U \to T_{g(x)} \mathbb{A}^n_Y$ is an isomorphism. The projection $\mathbb{A}^n_Y \to Y$ induces a surjection $T_{g(x)}\mathbb{A}^n_Y \to T_{f(x)} Y$. The composition is the surjection you want.