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