If \(y = \log_e\left(\displaystyle\frac{e^2}{x^2}\right)\) for \(x \ne 0\), then \(\displaystyle\frac{d^2y}{dx^2}\) equals
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{2}{x^2}\)
By the laws of logarithms, for \(x \ne 0\):
\(y = \log_e e^2 - \log_e x^2\)
\(= 2 - 2\log_e|x|\)
Since the derivative of \(\log_e|x|\) is \(\frac{1}{x}\):
\(\displaystyle \frac{dy}{dx} = -\frac{2}{x}\)
\(\displaystyle \frac{d^2y}{dx^2} = \frac{2}{x^2}\)