Let $M$ be a smooth manifold and $N$ a submanifold of M. Let $X_1,..,X_k\in\Gamma(TM)$ be vector fields on $M$, which restrict to vectorfields on $N$, i.e. for $n\in N$ it holds $X_{i,n}\in T_n N\subseteq T_nM$ for all $i$ and $n\in N$. Then $X_1|_N,...,X_k|_N\in \Gamma(TN)$ are vector fields on $N$.
Let now $f\in C^\infty(M)$ be a smooth function. Then we get two smooth functions on $N$: Restricting the derivative $X_1...X_k(f)$ on $M$ to $N$ and taking the derivative of $f|_N$ in $N$. Do these two functions coincide, i.e. does $$(X_1...X_k(f))|_N=X_1|_N...X_k|_N(f|_N)$$ always hold?
I seem to have answered the question by myself in the meanwhile. I would be happy if someone takes a look at my proof and comments.
Suppose we have shown that the claim holds for a single vectorfield $X$ tangent to $N$, i.e. $$X(f)|_N=X|_N(f|_N)$$ holds. Then we get the claim by inductio, since $$(X_1...X_k(f))|_N=X_1(X_2...X_k(f))|_N=X_1|_N(X_2...X_k(f)|_N)=...=X_1|_N...X_k|_N(f|_N).$$
So it leaves to prove the claim for a single vectorfield $X$ tangent to $N$. Let $n\in N$, since $X_n\in T_nN\subseteq T_nM$, there is a smooth curve $\gamma\colon (-\epsilon,\epsilon)\rightarrow N$, such that $\gamma'(0)=X_n$. Now it holds $$X(f)(n)=X_n(f)=\gamma'(0)(f)=(f\circ\gamma)'(0)=X_n(f|_N)=X|_N(f|_N)(n),$$ since $f$ depends only on the values of the curve $\gamma$ which are in $N$.