A) \[log\text{ }x\]
B) \[\log {{e}^{x}}\]
C) \[{{x}^{x}}log\text{ }x\]
D) \[{{x}^{x}}\log e\,x\]
E) \[{{x}^{x}}\log \text{ }(1-x)\]
Correct Answer: D
Solution :
Let \[y={{x}^{x}}\] \[\Rightarrow \]\[log\text{ }y=x\text{ }log\text{ }x\] On differentiating w. r. t. \[x,\]we get \[\frac{1}{y}\frac{dy}{dx}=x.\frac{1}{x}+\log x\] \[\Rightarrow \] \[\frac{dy}{dx}={{x}^{x}}(1+\log x)\] \[={{x}^{x}}(\log \,ex)\]
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