A) \[{{x}^{\ln x}}\ln x\]
B) \[{{x}^{\text{ln}\,x-1}}\text{ln}\,x\]
C) \[2{{x}^{\ln x-1}}\ln \,x\]
D) \[{{x}^{\ln x-2}}\]
Correct Answer: C
Solution :
\[y={{x}^{\ln x}}\] Þ \[\ln y={{(\ln x)}^{2}}\]Þ \[\frac{1}{y}\frac{dy}{dx}=\frac{2\ln x}{x}\] Þ \[\frac{dy}{dx}=y\frac{2\ln x}{x}=\frac{2({{x}^{\ln x}})\ln x}{x}\]Þ \[\frac{dy}{dx}=2{{x}^{\ln x-1}}\ln x\].
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