A) \[2{{x}^{({{\log }_{e}}x-1)}}.{{\log }_{e}}x\]
B) \[{{x}^{({{\log }_{e}}x-1)}}\]
C) \[\frac{2}{x}{{\log }_{e}}x\]
D) \[{{x}^{({{\log }_{e}}x-1)}}.{{\log }_{e}}x\]
Correct Answer: A
Solution :
Let\[y={{x}^{{{\log }_{e}}x}}\]Þ \[{{\log }_{e}}y={{\log }_{e}}x{{\log }_{e}}x={{({{\log }_{e}}x)}^{2}}\] Þ \[\frac{1}{y}\frac{dy}{dx}=2{{\log }_{e}}x.\frac{1}{x}\] \[\therefore \frac{dy}{dx}=2{{x}^{({{\log }_{e}}x-1)}}{{\log }_{e}}x\].
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