Let $f\in C^\infty(\mathbb R;\mathbb R)$ such that $f(0)=0$ and $D f(0)=0$. Naively I believe that
the $k$-th integer power $f^k$ of $f$, behaving like $x^{2k}$ near $0$ satisfies
$D^r f^k(0) = 0 $ if $r<2k$
but I have some problems to prove it rigorously. Can anybody help me? Should I use Faa di Bruno' formula or can I avoid it and use some simpler argument? Thanks in advance.