Question

If $f\left(x\right)={\log }_5{\log }_{3}x,$ then $f^{\prime }\left(e\right)$ is equal to

• A. $e{\log }_{e}5$
• B. $e{\log }_{e}3$
• C. $\frac{1}{e{\log }_{e}5}$
• D. $\frac{1}{e{\log }_{e}3}$

Answer 1

The correct option is C $\frac{1}{e{\log }_{e}5}$

$f\left(x\right)={\log }_5{\log }_{3}x$
We know
${\log }_{a}x=\frac{{\log }_{e}x}{{\log }_{e}a},$
where $a$ is a constant.
and hence, $\frac{d}{dx}\left({\log }_{a}x\right)=\frac{1}{x{\log }_{e}a}$
So, $f^{\prime }\left(x\right)=\frac{1}{{\log }_{3}x\cdot {\log }_{e}5}\times \frac{d}{dx}\left({\log }_{3}x\right)$
$=\frac{1}{{\log }_{3}x\cdot {\log }_{e}5}\times \frac{1}{x{\log }_{e}3}$
$\therefore f^{\prime }\left(e\right)=\frac{1}{e{\log }_{e}5}$