Question

Derivative of the function $f(x)=lo{g}_5(lo{g}_{7}x)$, $x>7$ is

• A. $\frac{1}{x\left(log5\right)\left(log7\right)\left(lo{g}_{7}x\right)}$
• B. $\frac{1}{x\left(log5\right)\left(log7\right)}$
• C. $\frac{1}{x\left(logx\right)}$
• D. None of these

Answer 1

Answer: (A)

$<br/>\frac{1}{x\log (5)\log (7){\log }_7(x)}<br/>$
Explanation for the correct option:
Step-1: Simplify the given data.
$<br/>f(x)={\log }_5\left({\log }_7(x)\right),x>7<br/>$
$<br/>\Rightarrow f(x)=\frac{{\log }_e\left({\log }_7(x)\right)}{{\log }_e(5)}\left(\because {\log }_b(a)=\frac{{\log }_e\left({}^a\right)}{{\log }_e(b)}\right)<br/>$
Step-2: Differentiate with respect to $x$
$\Rightarrow f^{\prime }(x)=\frac{1}{\log (5)}\times \frac{d}{dx}\left({\log }_e\left({\log }_7(x)\right)\right)$
$\Rightarrow f\prime (x)=\frac{1}{\log (5)}\times \frac{1}{{\log }_7(x)}\times \frac{d}{dx}\left({\log }_7(x)\right)$
$\Rightarrow f\prime (x)=\frac{1}{\log (5)}\times \frac{1}{{\log }_7(x)}\times \frac{d}{dx}\left(\frac{\log (x)}{\log (7)}\right)$
$\Rightarrow f\prime (x)=\frac{1}{\log (5)}\times \frac{1}{{\log }_7(x)}\times \frac{1}{\log (7)}\times \frac{1}{x}$
$\Rightarrow f^{\prime }(x)=\frac{1}{x\log (5)\log (7){\log }_7(x)}$