If $$y = log_{√e}(sin x)$$ then $$\frac{dy}{dx}$$ is :​

Correct option is C

Given:
​$$y = \log_{\sqrt{e}}(\sin x)$$​​
Concept used:
$$\log_a b = \frac{\ln b}{\ln a}, and \frac{d}{dx}(\ln(\sin x)) = \cot x$$​​
Solution:
​$$y = \frac{\ln(\sin x)}{\ln(\sqrt{e})} \\ \ \\\ln(\sqrt{e}) = \ln(e^{1/2}) = \frac{1}{2}$$​​
So,
​$$y = \frac{\ln(\sin x)}{1/2} = 2\ln(\sin x)$$​​
Differentiate:
​$$\frac{dy}{dx} = 2 \times \frac{\cos x}{\sin x} = 2\cot x$$​​
Correct answer is (c) $$2 \cot x \\$$​​