If f(x)= _e( _ex), then write the value of f'(e). - Vidyadip

Question

If $\text{f(x)}=\log_\text{e}(\log_\text{e}\text{x}),$ then write the value of f'(e).

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Answer

We have, $\text{f(x)}=\log_\text{e}(\log_\text{e}\text{x})$
Differentiating with respect to x,
$\text{f}'\text{(x)}=\frac{1}{\log_\text{e}\text{x}}\frac{\text{d}}{\text{dx}}(\log_\text{e}\text{x})$
$\Rightarrow\text{f}'\text{(x)}=\frac{1}{\log_\text{e}\text{x}}\Big(\frac{1}{\text{x}}\Big)$
$\Rightarrow\text{f}'\text{(e)}=\frac{1}{\log_\text{e}\text{e}}\Big(\frac{1}{\text{e}}\Big) \big[\because\text{x}=\text{e}\big]$
$\Rightarrow\text{f}'\text{(e)}=\frac{1}{\text{x}}\big[\because\log_\text{e}=1\big]$

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