If \(y=\log\left(\displaystyle\frac{x^5}{e^5}\right)\), then \(\displaystyle\frac{d^2y}{dx^2}\) is,
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{-5}{x^2}\)
By the laws of logarithms:
\(y = \log x^5 - \log e^5\)
\(y = 5\log x - 5\)
Differentiating twice:
\( \displaystyle \frac{dy}{dx} = \frac{5}{x} \)
\( \displaystyle \frac{d^2y}{dx^2} = -\frac{5}{x^2} \)