How to show this map between Zariski tangent spaces is injective

Question

Let $f: X \rightarrow Y$ be a flat morphism of finite type of schemes and let $Z$ be a closed subscheme of $Y$. I want to consider $X \times_{Y} Z$. Let $x \in X \times_Y Z$ such that $f(x) \in Z$. Let $\pi: X \times_Y Z \rightarrow X$. This induces a map between the tangent spaces $$ T_x X \times_Y Z \rightarrow T_{\pi(x)}X \otimes_{k(\pi(x))}k(x). $$ $k(x)$ is the residue field of $x$. I was wondering how can I deduce that this map is injective? Any comments would be appreciated. Thank you.