Finding the kernel of Pushforward of $f:\mathbb R^n\rightarrow \mathbb R^k$
I'm trying to show that $T_{p}M \subset ker(f_{*})$. We defined $T_{p}M = \{(p,m)| m \in M\}$ and $f_{*}(p,x)=(f(p),Df(p)(x))$. I've reduced the problem to showing that $Df(p)(x)=0$ for all $x$ in $M$. I can't seem to go further. How do I use the fact the f is a submersion?
EDIT: Ah, I was using an incorrect definition of the tangent space of manifold.