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

To find the derivative of the function \( f(x) = \log_5(\log_7 x) \), where \( x>7 \), we will employ the chain rule and the properties of logarithms. Let's break down the process step-by-step.

Step 1: Understand the Function

The function can be rewritten using the change of base formula as:

\(f(x) = \frac{\log (\log_7 x)}{\log 5}\)

Step 2: Differentiate \( \log_7 x \)

Using the change of base formula, we express \( \log_7 x \) in terms of natural logarithms:

\(\log_7 x = \frac{\log x}{\log 7}\)

Differentiating with respect to \( x \), we get:

\(\frac{d}{dx}\left(\log_7 x\right) = \frac{d}{dx}\left(\frac{\log x}{\log 7}\right) = \frac{1}{x \log 7}\)

Step 3: Differentiate \( f(x) = \frac{\log (\log_7 x)}{\log 5} \)

Applying the chain rule to differentiate \( \log(\log_7 x) \), we have:

\(\frac{d}{dx} [\log(\log_7 x)] = \frac{1}{\log_7 x} \cdot \frac{d}{dx}[\log_7 x]\)

Substituting the derivative from Step 2, we get:

\(\frac{1}{\log_7 x} \cdot \frac{1}{x \log 7} = \frac{1}{x (\log 7) (\log_7 x)}\)

Step 4: Differentiate \( f(x) = \frac{\log(\log_7 x)}{\log 5} \)

Substituting the result from Step 3, we find:

\(f'(x) = \frac{1}{\log 5} \cdot \frac{1}{x (\log 7) (\log_7 x)}\)

This simplifies to:

\(f'(x) = \frac{1}{x (\log 5) (\log 7) (\log_7 x)}\)

Conclusion

The correct derivative of the function \( f(x) = \log_5(\log_7 x) \) is:

\(\frac{1}{x(\log 5)(\log 7)(\log_7 x)}\)

Thus, the correct option is: \(\frac{1}{x(\log 5)(\log 7)(\log_7 x)}\)