Question

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

• A. $\frac{2}{6}^{-\frac{\pi}{6}}[6 \log 2-\sqrt{3} \pi]$
• B. $2^{\frac{\pi}{6}}[6 \log 2+\sqrt{3} \pi]$
• C. $\frac{2^{-\frac{\pi}{6}}}{6}[6 \log 2+\sqrt{3} \pi]$
• D. None of the above

Answer 1

Answer: (A)

Given $y=|\sin x|^{|x|}$
In the neighbourhood of $-\frac{\pi}{6},|x|$ and $|\sin x|$ both are negative
i.e., $y=(-\sin x)^{-x}$
Taking log on both sides, we get
$\log y=-x \log (-\sin x)$
$\Rightarrow \frac{1}{y} \frac{d y}{d x}=(-x) \frac{1}{-\sin x}(-\cos x)+\log (-\sin x)(-1)$
$=-x \cdot \cot x-\log (-\sin x)$
$=-[x \cot x+\log (-\sin x)]$
$\Rightarrow \frac{d y}{d x}=-y[x \cot x+\log (-\sin x)]$
$\therefore \left(\frac{d y}{d x}\right)_{x=-\frac{\pi}{6}}=(2)^{-\frac{\pi}{6}} \frac{[6 \log 2-\sqrt{3} \pi]}{6}$