Question

If \( y = |\sin x|^{|x|} \), then the value of \( \frac{dy}{dx} \) at \( x = -\frac{\pi}{6} \) is

• A. $2^{-\pi/6}(6\log 2 - \sqrt{3}\pi)$
• B. $2^{\pi/6}(6\log 2 + \sqrt{3}\pi)$
• C. $2^{-\pi/6}(6\log 2 + \sqrt{3}\pi)$
• D. None of these

Hint:

For $f(x)^{g(x)}$, always use logarithmic differentiation.

Answer 1

The Correct Option is A

Concept: Logarithmic differentiation for functions of type $y = f(x)^{g(x)}$.

Step 1: Simplify absolute values.
At $x = -\frac{\pi}{6}$: \[ |x| = \frac{\pi}{6}, \quad |\sin x| = \sin\left(\frac{\pi}{6}\right)=\frac{1}{2} \] \[ y = \left(\frac{1}{2}\right)^{\pi/6} \]

Step 2: Take logarithm.
\[ \ln y = |x| \ln|\sin x| \]

Step 3: Differentiate.
\[ \frac{1}{y}\frac{dy}{dx} = \frac{d}{dx}(|x|)\ln|\sin x| + |x|\frac{\cos x}{\sin x} \]

Step 4: Substitute values.
\[ \frac{d}{dx}(|x|) = -1 \text{ (since } x<0) \] \[ \ln\left(\frac{1}{2}\right) = -\ln 2 \] \[ \cot\left(-\frac{\pi}{6}\right) = -\sqrt{3} \]

Step 5: Compute derivative.
\[ \frac{dy}{dx} = y[6\ln 2 - \sqrt{3}\pi] \] \[ = 2^{-\pi/6}(6\log 2 - \sqrt{3}\pi) \] Conclusion:
Answer = $2^{-\pi/6}(6\log 2 - \sqrt{3}\pi)$