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If $\mathrm{y}=|\sin \mathrm{x}| \mathrm{x} \mid$, then what is the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{\pi}{6} ?$
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$\frac{2^{-\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}$
$\begin{aligned} y &=(-\sin x)^{-x} \\ & \frac{d y}{d x}=(-\sin )^{-x}\left[\frac{x}{\sin x}(-\cos x)+\log (-\sin x) \cdot(-1)\right] \\ &\left[\frac{d y}{d x}\right] x=-\frac{\pi}{6}=\left(\frac{1}{2}\right)^{\pi / 6}\left[-\frac{\pi}{6} \sqrt{3}-\log \frac{1}{2}\right] \\ &=2^{-\pi / 6}\left(\frac{6 \log 2-\sqrt{3} \pi}{6}\right) \end{aligned}$
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