Question

Derivative of the function $f(x) = log_5(log_7x)$, $x > 7$ is

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

Answer 1

Answer: (A)

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