If $\mathrm{y}=\left(\log _{\mathrm{x}} \sin \mathrm{x}\right)^{\mathrm{x}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}=$

• A

  $y\left[\frac{x \sin x}{\log \cos x}+\log (\log \sin x)+\frac{1}{\log x}-\log (\log x)\right]$
• B

  $y\left[\frac{x \cos x}{\log \sin x}-\log (\log \sin x)+\frac{1}{\log x}+\log (\log x)\right]$
• Correct answer
  C

  $y\left[\frac{x \cot x}{\log \sin x}+\log (\log \sin x)-\frac{1}{\log x}-\log (\log x)\right]$
• D

  $y\left[\frac{x \cot x}{\log \sin x}-\log (\log \sin x)+\frac{1}{\log x}-\log x\right]$

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