How to show that $\ker d\pi_x=T_x(\pi^{-1}(\pi(x)))$?

Question

If $\pi:M\to N$ be a submersion then how to show that $\ker d\pi_x=T_x(\pi^{-1}(\pi(x)))$?

I have another question about submersions:

It is true that $N$ can be thought as submanifold of $M$?

Answer 1

Hint: $\pi{\big|_{\pi^{-1}(\pi(x))}}$ is a constant smooth map that sends all its domain to $\pi(x)$, for all $x\in M$.