A) 1
B) e
C) \[\frac{1}{2e}\]
D) 0
Correct Answer: C
Solution :
Given \[f(x)={{\log }_{{{x}^{2}}}}({{\log }_{e}}x)=\frac{1}{2}{{\log }_{x}}({{\log }_{e}}x)\] \[\Rightarrow \] \[f(x)=\frac{1}{2}\frac{{{\log }_{e}}{{\log }_{e}}x}{{{\log }_{e}}x}\] On differentiating both sides w. r. t x, we get \[f'(x)=\frac{1}{2}\frac{{{\log }_{e}}x\left( \frac{1}{x{{\log }_{e}}x} \right)-{{\log }_{e}}{{\log }_{e}}x\times \frac{1}{x}}{{{({{\log }_{e}}x)}^{2}}}\] \[\Rightarrow f'(x)=\frac{1}{2}\frac{\left[ \frac{1}{x}-\frac{1}{x}{{\log }_{e}}{{\log }_{e}}x \right]}{{{({{\log }_{e}}x)}^{2}}}\] At \[x=e,f'(e)=\frac{1}{2}\frac{\left[ \frac{1}{e}-\frac{1}{e}{{\log }_{e}}1 \right]}{{{1}^{2}}}\] \[\Rightarrow \] \[f'(e)=\frac{1}{2e}\]
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