Q.If $f(x) = \log_x (\log_e x)$, then the value of $f'(e)$ is

Convert to natural logs, differentiate with the quotient rule, and evaluate at $x=e$ where $\ln x=1,\ \ln(\ln x)=0$.

Differentiating a change-of-base logarithm is a CBSE/NCERT Class 12 continuity and differentiability exercise.

Using change of base, $f(x)=\log_x(\log_e x) = \dfrac{\ln(\ln x)}{\ln x}$.

Let $N=\ln(\ln x)$ and $D=\ln x$. Then

$N' = \frac{1}{\ln x}\cdot\frac1x = \frac{1}{x\ln x}, \qquad D' = \frac1x.$

Quotient rule: …