Question

If $y=|\sin x{|}^{|x|}$, then the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{6}$.

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

Answer 1

Answer: (A)

${\frac{2}{6}}^{-\frac{\pi }{6}}[6\log 2-\sqrt{3}x]$

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
$\log y=-x\log (-\sin x)$
$\Rightarrow \frac{1}{y}\frac{dy}{dx}=(-x)\frac{1}{-\sin x}(-\cos x)+\log (-\sin x)(-1)$
$=-x\cdot \cot x-\log (-\sin x)$
$=-[x\cot x+\log (-\sin x\left)\right]$
$\Rightarrow \frac{dy}{dx}=-y[x\cot x+\log (-sinx\left)\right]$
Therefore, ${\left(\frac{dy}{dx}\right)}_{x=-\frac{\pi }{6}}=(2{)}^{-\frac{\pi }{6}}\frac{[6\log 2-\sqrt{3}\pi ]}{6}$.