Question

The derivative of \(log_8(log_5x)\) w. r. t. \(x\) is

• A. \(\frac{1}{log_58logx}\)
• B. \(\frac{1}{xlog5logx}\)
• C. \(\frac{1}{xlogxlog8log5}\)
• D. \(\frac{1}{xlog8logx}\)

Hint:

Convert to natural logs and use the chain rule.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Change of base: \(\log_8 u = \dfrac{\log u}{\log 8}\) and \(\log_5 x = \dfrac{\log x}{\log 5}\).

Step 2: Write the function:
\[ y = \frac{1}{\log 8}\,\log\left(\frac{\log x}{\log 5}\right) \]

Step 3: Differentiate:
The derivative of \(\log u\) is \(u'/u\). Here \(u = \dfrac{\log x}{\log 5}\), so \(u' = \dfrac{1}{x\log5}\).
\[ \frac{dy}{dx} = \frac{1}{\log8}\cdot\frac{1/(x\log5)}{(\log x)/\log5} = \frac{1}{x\log8\,\log x} \]

Step 4: Match:
The \(\log 5\) cancels completely, so option (D) is the answer. Options (A), (B), (C) keep a stray \(\log5\) factor.

Final Answer:
The log 5 factor cancels, leaving 1/(x log 8 log x). \[ \boxed{\text{(D) }\dfrac{1}{x\log 8\,\log x}} \]