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: Outer then inner:
$\dfrac{d}{dx}\log_8 u = \dfrac{1}{u\log 8}\cdot u'$ with $u = \log_5 x$.

Step 2: Inner derivative:
$u' = \dfrac{1}{x\log5}$ and $u = \dfrac{\log x}{\log5}$, so $\dfrac{u'}{u} = \dfrac{1}{x\log x}$.

Step 3: Combine:
$\dfrac{dy}{dx} = \dfrac{1}{\log8}\cdot\dfrac{1}{x\log x}$, option (D).

Final Answer:
The derivative is 1/(x log 8 log x). \[ \boxed{\text{(D) }\dfrac{1}{x\log 8\,\log x}} \]