I'm reading in do Carmo's book about surfaces and in one definition it is said that a point $p$ is critical if the differential in that point is not surjective. Also, it is known in $\mathbb{R}$ that a point $p$ is critical if $f'(p)=0$.
My question:
Which is the relation between surjective and derivative?
The differential in one dimension is quite simple: if $f\colon \mathbb R\to \mathbb R$ is a differentiable function, then:
$$\mathrm d_a f\colon h\mapsto f'(a)h.$$
So you can see that $f'(p)=0$ if, and only if $\mathrm d_p f$ is not surjective (since it's image is the point $\{0\}$, and not $\mathbb R$).