If f ( x ) = log x 2 ( log x ) , then f ' ( e ) is equal to __________

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Question: Answered & Verified by Expert

If

f (x) = \log_{x^{2}} (\log x),

then

f^{'} (e)

is equal to __________

MathematicsDifferentiationAP EAMCETAP EAMCET 2020 (23 Sep Shift 1)

Options:

A $e$
B $\frac{1}{e}$
C $\frac{1}{2 e}$
D $2 e$

Solution:

2293 Upvotes Verified Answer

The correct answer is:

\frac{1}{2 e}

$f (x) = \log_{x^{2}} (\log x)$

$\Rightarrow f (x) = \frac{\ln (\ln x)}{\ln x^{2}}$

$\Rightarrow f' (x) = \frac{(\ln x^{2}) (\frac{1}{x \ln x}) - (\ln (\ln x)) (\frac{2}{x})}{(2 \ln x)^{2}}$

$\Rightarrow f' (e) = \frac{\frac{2}{e}}{4}$

$= \frac{1}{2 e}$

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