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 $df_x:T_x(X) \to T_{f(x)}(Y)$.
("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.?)
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.
Your idea doesn't make sense as written, since $df_x$ is defined on $T_x(X)$, not $X$ itself, and so $df_x(x)$ makes no sense (and neither does $df_w(w)$).
To answer this question, you will need to have a much stronger understanding of basic ideas like derivatives (i.e. the maps $df_x$) and tangent spaces than you currently seem to. Perhaps you should try some more basic questions first.
Also, this implies the result about tangent spaces of intersections pretty immediately: to say that submanifolds $X$ and $Y$ of $Z$ are transverse is the same as saying the the inclusion $\iota:X \to Z$ is transverse to $Y$. The preimage of $Y$ under $\iota$ is precisely $X \cap Y$. Now just apply the general statement in this particular case.