A) \[1\]
B) \[\frac{1}{e}\]
C) \[\frac{1}{2e}\]
D) \[0\]
Correct Answer: C
Solution :
Given, \[f(x)={{\log }_{{{x}^{2}}}}({{\log }_{e}}x)=\frac{1}{2}{{\log }_{x}}({{\log }_{e}}x)\] \[\Rightarrow \] \[f(x)=\frac{1}{2}\frac{{{\log }_{e}}\,{{\log }_{e}}x}{{{\log }_{e}}x}\] \[\Rightarrow \]\[f'(x)=\frac{1}{2}\frac{{{\log }_{e}}x\left[ \frac{1}{x\,{{\log }_{e}}x} \right]-{{\log }_{e}}{{\log }_{e}}x\times \frac{1}{x}}{{{({{\log }_{e}}x)}^{2}}}\] \[\Rightarrow \] \[f'(x)=\frac{1}{2}\frac{\frac{1}{x}-\frac{1}{x}{{\log }_{ex{{)}^{{}}}}}{{\log }_{e}}x}{{{({{\log }_{e}})}^{2}}}\] At \[x=e,\] \[f'(e)=\frac{1}{2}\frac{\frac{1}{e}-\frac{1}{e}{{\log }_{e}}1}{{{(1)}^{2}}}\] \[\Rightarrow \] \[f'(e)=\frac{1}{2e}\]
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