A) \[{{(\sin x)}^{\log x}}\left[ \frac{1}{x}\log \sin x+\cot x \right]\]
B) \[{{(\sin x)}^{\log x}}\left[ \frac{1}{x}\log \sin x+\cot x\log x \right]\]
C) \[{{(\sin x)}^{\log x}}\left[ \frac{1}{x}\log \sin x+\log x \right]\]
D) None of these
Correct Answer: B
Solution :
Let \[y={{(\sin x)}^{\log x}}\Rightarrow {{\log }_{e}}y={{\log }_{e}}x{{\log }_{e}}\sin x\] Þ \[\frac{dy}{dx}={{(\sin x)}^{{{\log }_{e}}x}}=\left[ \frac{1}{x}{{\log }_{e}}\sin x+\cot x{{\log }_{e}}x \right]\].
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