Question

Derivative of $log_{e^{2}} \left(log\,x\right)$ with respect to $x$ is $\ldots\ldots$

• A. $\frac{2}{x\, log x}$
• B. $\frac{1}{x\, log x}$
• C. $\frac{1}{x\, log x^{2}}$
• D. $\frac{2}{log\,x}$

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
$\frac{d y}{d x} =\frac{1}{2} \frac{1}{\log x} \frac{d}{d x}(\log x)$
$=\frac{1}{\log x^{2}} \cdot \frac{1}{x}=\frac{1}{x \log x^{2}}$