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Question
If f (x) = loge (loge x), then write the value of `f' (e)` ?
Solution
\[\text { We have }, f\left( x \right) = \log_e \left( \log_e x \right)\]
Differentiating with respect to x,
\[f'\left( x \right) = \frac{1}{\log_e x}\frac{d}{dx}\left( \log_e x \right) \]
\[ \Rightarrow f'\left( x \right) = \frac{1}{\log_e x}\left( \frac{1}{x} \right)\]
\[ \Rightarrow f'\left( e \right) = \frac{1}{\log_e e}\left( \frac{1}{e} \right) \left[ \because x = e \right]\]
\[ \Rightarrow f'\left( e \right) = \frac{1}{e} \left[ \because \log_e e = 1 \right]\]
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