Question

Derivative of $lo{g}_{e^2}\left(logx\right)$ with respect to $x$ is $\dots \dots$

• A. $\frac{2}{xlogx}$
• B. $\frac{1}{xlogx}$
• C. $\frac{1}{xlog{x}^2}$
• D. $\frac{2}{logx}$

Answer 1

Answer: (C)

Let $y={\log }_{e^2}(\log x)$
$\Rightarrow y=\frac{1}{2}{\log }_e(\log x)$
$\left(\because {\log }_{a^n}(x)=\frac{1}{n}{\log }_{a}x\right)$
On differentiating both sides w.r.t. $x,$ we get
$\frac{dy}{dx}=\frac{1}{2}\frac{1}{\log x}\frac{d}{dx}(\log x)$
$=\frac{1}{\log x^2}\cdot \frac{1}{x}=\frac{1}{x\log x^2}$