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