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Question: If f(x) = log<sub>x</sub> (ln (x)) then f '(x) at x = e is...

If f(x) = logx (ln (x)) then f '(x) at x = e is

A

0

B

1

C

e

D

1/e

Answer

1/e

Explanation

Solution

f(x) = ln(ln(x))ln(x)\frac{\mathcal{l}n(\mathcal{l}n(x))}{\mathcal{l}n(x)}

f '(x) = 1lnx×1x×ln(x)−ln(ln(x))×1/x[ln(x)]2\frac{\frac{1}{\mathcal{l}nx} \times \frac{1}{x}\mathcal{\times l}n(x)\mathcal{- l}n(\mathcal{l}n(x)) \times 1/x}{\lbrack\mathcal{l}n(x)\rbrack^{2}}

f '(x) = 1e−01\frac{\frac{1}{e} - 0}{1} = 1/e