Smooth bijection and tangent spaces

Question

Let $f:M\to N$ be a smooth bijection between manifolds with same dimension. Do we necessarily have

$$df_p(T_pM)=T_{f(p)}N.$$

I think it is probably not true. But I can't give a counterexample...

Answer 1

The map $f:\mathbb{R}\to\mathbb{R}$ given by $f:x\mapsto x^3$ is a smooth bijection but $df_0(T_0\mathbb{R}) = 0\neq T_{f(0)}\mathbb{R}$.

Answer 2

How about $f: \mathbb{R} \to \mathbb{R}$ by $f(x)=x^3$? Certainly smooth, certainly bijective. However, $df_0$ will map the $1$-dimensional tangent space to a point. The root issue is that a smooth bijection can still have critical points.