If \( y=\log_{3}(\log x) \), (Consider \( \log_{e}x=\log x \) ) then the value of \( \displaystyle\frac{dy}{dx} \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{x\log 3\log x} \)
Changing the base to e:
\( \displaystyle y=\log_3(\log x)=\frac{\log(\log x)}{\log 3} \)
By the chain rule, with \( \log 3 \) a constant:
\( \displaystyle \frac{dy}{dx}=\frac{1}{\log 3}\cdot\frac{1}{\log x}\cdot\frac{1}{x} \)
\( \displaystyle =\frac{1}{x\log 3\log x} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{\log x} \) — Incorrect. Both factors \( \frac{1}{x} \) (derivative of log x) and \( \frac{1}{\log 3} \) (change of base) are missing.
Option (3) \( \displaystyle\frac{1}{\log 3\log x} \) — Incorrect. The factor \( \frac{1}{x} \) from differentiating the inner function log x is missing.
Option (4) \( \displaystyle\frac{1}{x\log x} \) — Incorrect. This is the derivative of \( \log(\log x) \); the change-of-base factor \( \frac{1}{\log 3} \) is missing.