A) \[\frac{1}{3e}(1-{{\log }_{e}}2)\]
B) \[\frac{1}{3e}(1+{{\log }_{e}}2)\]
C) \[\frac{1}{3e}(-1+{{\log }_{e}}2)\]
D) \[-\frac{1}{3e}(1+{{\log }_{e}}2)\]
E) \[\frac{1}{3e}({{\log }_{e}}2)\]
Correct Answer: A
Solution :
\[\because \]\[f(x)={{\log }_{{{x}^{3}}}}(\log {{x}^{2}})=\frac{\log (2\log x)}{3\log x}\] On differentiating w.r.t.\[x,\]we get \[f(x)=\frac{\log x.\frac{1}{2\log x}.\frac{2}{x}-\log (2\log x)\frac{1}{x}}{3{{(\log x)}^{2}}}\] \[\Rightarrow \]\[f(e)=\frac{1}{3e}\left\{ \frac{1-\log 2}{{{(1)}^{2}}} \right\}=\frac{1}{3e}(1-\log 2)\]
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