If $f(x)=log _{e}left(log _{e} x ight)$, then write the value of $f^{prime}(e)$.

Question:

If $f(x)=\log _{e}\left(\log _{e} x\right)$, then write the value of $f^{\prime}(e)$.

Solution:

$f(x)=\log _{e}\left(\log _{e} x\right)$

Using the Chain Rule of Differentiation,

$f(x)=\frac{1}{\log _{e} x} \cdot \frac{1}{x}$

So, $f(e)=\frac{1}{\log _{e} e} \cdot \frac{1}{e}$

$=\frac{1}{e}$ (Ans)

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