A) \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\]
B) \[{{x}^{n-2}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\]
C) \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n-\log x]\]
D) None of these
Correct Answer: A
Solution :
\[y={{x}^{n}}\log x+x{{(\log x)}^{n}}\] \[\frac{dy}{dx}=n{{x}^{n-1}}\log x+{{x}^{n}}.\left( \frac{1}{x} \right)+xn{{(\log x)}^{n-1}}.\left( \frac{1}{x} \right)+1.{{(\log x)}^{n}}\] \[={{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\].
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