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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.

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    @QiaochuYuan: I'm bit ashamed I didn't look at the easiest thing to do. thanks for opening my eyes again with this hint. I accept it as an answer if you post your comment as such.2012-08-12

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