Question

Derivative of the function $f\left(x\right)={\log }_5\left({\log }_7\left(x\right)\right),x>7$is

• A. $\frac{1}{x\log \left(5\right)\log \left(7\right){\log }_7\left(x\right)}$
• B. $\frac{1}{x\log \left(5\right)\log \left(7\right)}$
• C. $\frac{1}{x\log \left(x\right)}$
• D. none of these

Answer 1

The correct option is A

$\frac{1}{x\log \left(5\right)\log \left(7\right){\log }_7\left(x\right)}$

Explanation for the correct option:

Step-1: Simplify the given data.

$f\left(x\right)={\log }_5\left({\log }_7\left(x\right)\right),x>7$

$\Rightarrow f\left(x\right)=\frac{{\log }_e\left({\log }_7\left(x\right)\right)}{{\log }_e\left(5\right)}\left(\because {\log }_b\left(a\right)=\frac{{\log }_e\left(a\right)}{{\log }_e\left(b\right)}\right)$

Step-2: Differentiate with respect to $x$

$$\begin{gathered} \Rightarrow f\text{'}\left(x\right)=\frac{1}{\log \left(5\right)}\times \frac{d}{dx}\left({\log }_e\left({\log }_7\left(x\right)\right)\right) \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{\log \left(5\right)}\times \frac{1}{{\log }_7\left(x\right)}\times \frac{d}{dx}\left({\log }_7\left(x\right)\right) \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{\log \left(5\right)}\times \frac{1}{{\log }_7\left(x\right)}\times \frac{d}{dx}\left(\frac{\log \left(x\right)}{\log \left(7\right)}\right) \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{\log \left(5\right)}\times \frac{1}{{\log }_7\left(x\right)}\times \frac{1}{\log \left(7\right)}\times \frac{1}{x} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{x\log \left(5\right)\log \left(7\right){\log }_7\left(x\right)} \end{gathered}$$

Hence, correct answer is option $\left(A\right)$