If $y = \log_e(\frac{e^2}{x^2})$ for $x≠0$, then $\frac{d^2y}{dx^2}$ equals

$\frac{2}{x^2}$

The correct answer is Option (3) → $\frac{2}{x^2}$

Given:

$y = \log_e\left(\frac{e^2}{x^2}\right)$

$\Rightarrow y = \log_e(e^2) - \log_e(x^2) = 2 - 2\log_e x$

Differentiate with respect to $x$:

$\frac{dy}{dx} = -2 \cdot \frac{1}{x} = -\frac{2}{x}$

Differentiate again:

$\frac{d^2y}{dx^2} = -2 \cdot \frac{d}{dx}\left(\frac{1}{x}\right) = -2 \cdot (-\frac{1}{x^2}) = \frac{2}{x^2}$

Hence, $\frac{d^2y}{dx^2} = \frac{2}{x^2}$