If `y=log_(e)((x^(2))/(e^(2)))`,then `(d^(2)y)/(dx^(2))` equals

Options

A. `(1)/(x)`
B. `(1)/(x^(2))`
C. `(2)/(x^(2))`
D. `(2)/(x^(2))`

Answer

To solve the problem, we need to find the second derivative of the function \( y = \log_e\left(\frac{x^2}{e^2}\right) \).

### Step-by-Step Solution:

1. **Rewrite the logarithmic expression:**
\[ y = \log_e\left(\frac{x^2}{e^2}\right) = \log_e(x^2) - \log_e(e^2) \]

2. **Apply the logarithmic properties:**
\[ y = 2\log_e(x) - 2 \]

3. **Differentiate \( y \) with respect to \( x \) to find the first derivative \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}(\log_e(x)) - 0 = 2 \cdot \frac{1}{x} = \frac{2}{x} \]

4. **Differentiate \( \frac{dy}{dx} \) to find the second derivative \( \frac{d^2y}{dx^2} \):**
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2}{x}\right) \]
Using the power rule:
\[ \frac{d^2y}{dx^2} = 2 \cdot \frac{d}{dx}(x^{-1}) = 2 \cdot (-1)x^{-2} = -\frac{2}{x^2} \]

### Final Answer:
\[ \frac{d^2y}{dx^2} = -\frac{2}{x^2} \]