Q245 / 270 · NDA 2018 (Phase 2) · Mathematics

If $y=|\sin x|^{|x|}$, then what is the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{6}$ ?

  • Correct answer
    A
    $\frac{2^{-\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}$
  • B
    $\frac{2^{\frac{\pi}{6}}(6 \ln 2+\sqrt{3} \pi)}{6}$
  • C
    $\frac{2^{-\frac{\pi}{6}}(6 \ln 2+\sqrt{3} \pi)}{6}$
  • D
    $\frac{2^{\frac{\pi}{6}}(6 \ln 2-\sqrt{3} \pi)}{6}$

Solution

$\begin{aligned} & y=(-\sin x)^{-x} \\ & \frac{d y}{d x}=(-\sin )^{-x}\left[\frac{x}{\sin x}(-\cos x)+\log (-\sin x) \cdot(-1)\right] \\ & {\left[\frac{d y}{d x}\right] x=-\frac{\pi}{6}=\left(\frac{1}{2}\right)^{\pi / 6}\left[-\frac{\pi}{6} \sqrt{3}-\log \frac{1}{2}\right]} \\ & =2^{-\pi / 6}\left(\frac{6 \log 2-\sqrt{3} \pi}{6}\right)\end{aligned}$

Hence, the correct answer is option: 'A'.

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