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If it helps, it ask the value for $n=100$ and $x=\pi/2$.

I can't do it by induction because it has too many factors and trying to use an equality for $\cos(2x)$ didn't helped.

I don't see the relation in the derivatives.

Hagen von Eitzen
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Luis
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  • You will probably want to express this using Taylor series. You are looking for $100!$ times the coefficient of $(x - \pi/2)^{100}$ in the Taylor expansion of $f$, which you can write down as a product of the three Taylor series of each subpiece, which you know very well. – davidlowryduda Oct 07 '16 at 03:07

2 Answers2

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First note that $$ \left(\frac{d}{dx}\right)^n(xg(x))=x\left(\frac{d}{dx}\right)^ng(x) +n\left(\frac{d}{dx}\right)^{n-1}g(x) $$ (you can check this by induction). Secondly your function can be written $$ f(x)=x\times\frac12\left(\sin(3x)-\sin(x)\right). $$ Thus $$ \left(\frac{d}{dx}\right)^{100}f(x) =\frac x2\left(3^{100}\sin(3x)-\sin(x)\right) +\frac{100}2\left(-3^{99}\cos(3x)+\cos(x)\right). $$ Now you can substitute $x=\frac{\pi}2$.

stewbasic
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Observe \begin{align} x\sin x \cos 2x =&\ x\frac{e^{ix}-e^{-ix}}{2i} \frac{e^{2ix}+e^{-2ix}}{2} = \frac{x}{4i}\left(e^{3ix}-e^{ix}+e^{-ix}-e^{-3ix} \right)\\ =&\ \frac{1}{2}x\sin 3x-\frac{1}{2}x\sin x \\ =&\ \frac{1}{2}\left(x-\frac{\pi}{2}\right)\sin 3x -\frac{1}{2}\left(x-\frac{\pi}{2}\right)\sin x + \frac{\pi}{4}\sin 3x -\frac{\pi}{4}\sin x\\ =&\ -\frac{1}{2}\left(x-\frac{\pi}{2}\right)\cos\left(3 \left(x-\frac{\pi}{2} \right) \right)- \frac{1}{2}\left(x-\frac{\pi}{2}\right)\cos \left(x-\frac{\pi}{2}\right) \\ &- \frac{\pi}{4}\cos\left(3 \left(x-\frac{\pi}{2} \right) \right)-\frac{\pi}{4}\cos\left(x-\frac{\pi}{2}\right). \end{align} Since \begin{align} \cos\left(3 \left(x-\frac{\pi}{2} \right) \right)=\sum^\infty_{k=0} (-1)^k\frac{3^{2k}(x-\pi/2)^{2k}}{(2k)!} \end{align} and \begin{align} \cos \left(x-\frac{\pi}{2} \right)=\sum^\infty_{k=0} (-1)^k\frac{(x-\pi/2)^{2k}}{(2k)!} \end{align} then it follows the \begin{align} f^{(100)}(\pi/2) = -\frac{\pi}{4}100! \left(\frac{1}{100!}+\frac{3^{100}}{100!} \right) = -\frac{\pi}{4}(3^{100}+1). \end{align}

Jacky Chong
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