Differentiation of \(\log[\log(\log x^5)]\) with respect to \(x\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{5}{x(\log x^5)\log(\log x^5)}\)
Let \(y = \log[\log(\log x^5)]\). By the chain rule, working from the outside in, each logarithm gives a factor of 1 over its argument:
\( \displaystyle \frac{dy}{dx} = \frac{1}{\log(\log x^5)} \cdot \frac{1}{\log x^5} \cdot \frac{1}{x^5} \cdot \frac{d}{dx}(x^5) \)
\( \displaystyle = \frac{1}{\log(\log x^5)} \cdot \frac{1}{\log x^5} \cdot \frac{5x^4}{x^5} \)
\( \displaystyle = \frac{5}{x(\log x^5)\log(\log x^5)} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{5}{x\log(\log x^5)}\) — Incorrect. It leaves out the factor \(\frac{1}{\log x^5}\), the derivative of the middle logarithm.
Option (3) \(\displaystyle\frac{5x^2}{(\log x^5)\log(\log x^5)}\) — Incorrect. The derivative has the factor \(\frac{5}{x}\), not \(5x^2\).
Option (4) \(\displaystyle\frac{5x^4}{(\log x^5)\log(\log x^5)}\) — Incorrect. It leaves out the factor \(\frac{1}{x^5}\), the derivative of the innermost logarithm, and keeps only \(\frac{d}{dx}(x^5) = 5x^4\).