Question

If $f\left(x\right)={\log }_e\left\{\log \right(x\left)\right\},$ then $f^{\prime }\left(x\right)$ at $x=e$ is :

• A. $e$
• B. $-e$
• C. $e^2$
• D. $e^{-1}$

Answer 1

The correct option is D $e^{-1}$

Given, $f\left(x\right)={\log }_e\left\{\log \right(x\left)\right\},$.

Now, $f^{\prime }\left(x\right)=\frac{1}{x{\log }_{e}x}$.

Again $f^{\prime }\left(e\right)=\frac{1}{e{\log }_{e}e}$

or, $f^{\prime }\left(e\right)=\frac{1}{e}$.