If \( y=\sqrt[5]{\log_{e}(5x^{2})} \), then \( \displaystyle\frac{dy}{dx}= \)
(Take \( \log_{e}x=\log x \))
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{2}{5x\left[\log(5x^{2})\right]^{\frac{4}{5}}} \)
Writing the fifth root as a power,
\( \displaystyle y=\left[\log(5x^{2})\right]^{1/5} \)
By the chain rule,
\( \displaystyle \frac{dy}{dx}=\frac{1}{5}\left[\log(5x^{2})\right]^{-4/5}\cdot\frac{d}{dx}\log(5x^{2}) \)
The derivative of the inner logarithm is
\( \displaystyle \frac{d}{dx}\log(5x^{2})=\frac{10x}{5x^{2}} \)
\( \displaystyle =\frac{2}{x} \)
Hence
\( \displaystyle \frac{dy}{dx}=\frac{1}{5}\cdot\frac{2}{x}\cdot\frac{1}{\left[\log(5x^{2})\right]^{4/5}} \)
\( \displaystyle =\frac{2}{5x\left[\log(5x^{2})\right]^{4/5}} \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{2}{5}\log(5x)^{2} \) — Incorrect. The derivative must contain the factor \( \left[\log(5x^{2})\right]^{-4/5} \) from the chain rule; this option has no such factor.
Option (3) \( \displaystyle\frac{1}{5}\log(5x)^{2} \) — Incorrect. This also lacks the factor \( \left[\log(5x^{2})\right]^{-4/5} \) and the factor \( \frac{2}{x} \).
Option (4) \( \displaystyle\frac{2}{5x^{2}\left[\log(5x^{2})\right]^{\frac{4}{5}}} \) — Incorrect. The derivative of \( \log(5x^{2}) \) is \( \frac{2}{x} \), not \( \frac{2}{x^{2}} \).