Question

The differential coefficient of $f\left({\log }_{e}x\right)$ with respect to $x$, where $f\left(x\right)={\log }_{e}x$ is

• A. $\frac{x}{{\log }_{e}x}$
• B. $\frac{1}{x}{\log }_{e}x$
• C. $\frac{1}{x{\log }_{e}x}$
• D. none of these

Answer 1

Answer: (C)

$\frac{1}{x{\log }_{e}x}$

We have to find derivative of $f\left({\log }_{e}x\right)$ w.r. to $x$
We have $f\left(x\right)={\log }_{e}x$
Therefore,
$\frac{df\left({\log }_{e}x\right)}{dx}=\frac{d}{dx}\left[{\log }_e\left({\log }_{e}x\right)\right]$
$=\frac{1}{{\log }_{e}x}\times \frac{d}{dx}\left({\log }_{e}x\right)$
$=\frac{1}{x.{\log }_{e}x}$