Question

Derivative of the function \(f(x) = \log_5(\log_7 x)\), \(x>7\) is

• A. \(\frac{1}{x(\log 5)(\log 7)(\log_7 x)}\)
• B. \(\frac{1}{x(\log 5)(\log 7)}\)
• C. \(\frac{1}{x(\log x)}\)
• D. None of the above

Hint:

\(\frac{d}{dx}\log_a u = \frac{1}{u \ln a} \cdot \frac{du}{dx}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use chain rule and change of base formula.
Step 2: Detailed Explanation:
\(f(x) = \frac{\ln(\ln x/\ln 7)}{\ln 5}\)
\(f'(x) = \frac{1}{\ln 5} \cdot \frac{1}{\ln x/\ln 7} \cdot \frac{1}{x} \cdot \frac{1}{\ln 7}\)
= \(\frac{1}{x \ln 5 \ln 7 (\ln x/\ln 7)} = \frac{1}{x \ln 5 \ln 7 \log_7 x}\)
Since \(\log_5 e = 1/\ln 5\), etc.
Step 3: Final Answer:
\(f'(x) = \frac{1}{x(\log 5)(\log 7)(\log_7 x)}\).