If \(y = \log_e\left(\displaystyle\frac{x^3}{e^3}\right)\), then \(\displaystyle\frac{d^2y}{dx^2}\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-\displaystyle\frac{3}{x^2}\)
By the laws of logarithms:
\(y = \log_e x^3 - \log_e e^3\)
\(= 3\log_e x - 3\)
Differentiating twice:
\( \displaystyle \frac{dy}{dx} = \frac{3}{x} \)
\( \displaystyle \frac{d^2y}{dx^2} = -\frac{3}{x^2} \)