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Question: If \(f(x) = \log_{x}(\log x),\) then \(f^{'}(x)\) at \(x = e\) is...

If f(x)=log⁡x(log⁡x),f(x) = \log_{x}(\log x), then f′(x)f^{'}(x) at x=ex = e is

A

e

B

1/e

C

1

D

None of these

Answer

1/e

Explanation

Solution

f(x)=log⁡x(log⁡x)=log⁡(log⁡x)log⁡xf(x) = \log_{x}(\log x) = \frac{\log(\log x)}{\log x} ⇒ f′(x)=1x−1xlog⁡(log⁡x)(log⁡x)2f^{'}(x) = \frac{\frac{1}{x} - \frac{1}{x}\log(\log x)}{(\log x)^{2}}

⇒ f′(e)=1e−01=1ef^{'}(e) = \frac{\frac{1}{e} - 0}{1} = \frac{1}{e}