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> More generally, let $f: X \to Y$ be a map transversal to a submanifold $Z$ in $Y$. Then $W = f^{-1}(Z)$ is a submanifold of $X$. Prove that $T\_x(W)$ is the preimage of $T\_{f(x)}(Z)$ under the linear map $df\_x:T\_x(X) \to T\_{f(x)}(Y)$.
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> ("The tangent space to the preimage of $Z$ is the preimage of the tangent space of $Z$.") (Why does this imply [The tangent space to the intersection is the intersection of the tangent spaces.](https://math.stackexchange.com/questions/439071/the-tangent-space-to-the-intersection-is-the-intersection-of-the-tangent-spaces)?) 
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I am very lost at this question. My idea is that to show

$$T\_x(W) = \{v \in T\_x(X)\;|\;df\_x(v) \in T\_{f(z)}(Z)\}.$$

Not sure if this is the right track, and I have no clue how to proceed.

Any ideas? Thanks.