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Let be $f:(0, \frac{\pi}{6})\rightarrow R$, $f(x)=\tan(x)\tan(2x)\tan(3x)$. I need to prove that $f^{(n)}(x)>0$ for any $0 < x < \frac{\pi}{6}$, where $f^{(n)}(x)$ is the nth derivative of $f(x)$.

It seems rather cumbersome to derive it.

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up vote 4 down vote accepted

Hint: The derivative of $\tan x$ is $1 + \tan^2 x$. This implies that your $n$th derivative will be a big polynomial in $\tan x$, $\tan 2x$ and $\tan 3x$ with only positive coefficients.

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thanks! I got the point. – user 1618033 Jun 12 '12 at 7:41
    
Great! You're welcome. – Hans Lundmark Jun 12 '12 at 7:52

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