A) \[1\]
B) \[2\]
C) \[0\]
D) \[1/e\]
Correct Answer: D
Solution :
We have, \[f(x)={{\log }_{x}}({{\log }_{e}}x)\] \[=\frac{{{\log }_{e}}{{\log }_{e}}x}{{{\log }_{e}}x}\] On differentiating w.r.t.\[~x,\] we get \[f'(x)=\frac{{{\log }_{e}}x\cdot \frac{1}{{{\log }_{e}}x}\cdot \frac{1}{x}-{{\log }_{e}}{{\log }_{e}}x\cdot \frac{1}{x}}{{{({{\log }_{e}}x)}^{2}}}\] \[=\frac{1-{{\log }_{e}}{{\log }_{e}}x}{x{{({{\log }_{e}}x)}^{2}}}\] At\[x=e\], \[f'(e)=\frac{1-{{\log }_{e}}{{\log }_{e}}e}{e{{({{\log }_{e}}e)}^{2}}}=\frac{1-0}{e}\] \[=\frac{1}{e}\]
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