A) \[y[{{x}^{x}}(\log ex).\log x+{{x}^{x}}]\]
B) \[y[{{x}^{x}}(\log ex).\log x+x]\]
C) \[y[{{x}^{x}}(\log ex).\log x+{{x}^{x-1}}]\]
D) \[y[{{x}^{x}}({{\log }_{e}}x).\log x+{{x}^{x-1}}]\]
Correct Answer: C
Solution :
\[y={{x}^{({{x}^{x}})}}\Rightarrow \log y={{x}^{x}}\log x\] Þ \[\frac{1}{y}\frac{dy}{dx}=\frac{dz}{dx}.\log x+\frac{1}{x}.z\] , (where \[{{x}^{x}}=z\]) \[\Rightarrow \frac{dy}{dx}={{x}^{({{x}^{x}})}}\left[ {{x}^{x}}(\log ex).\log x+{{x}^{x-1}} \right]\],\[\left\{ \because \frac{dz}{dx}={{x}^{x}}\log ex \right\}\].
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