If `y=log((x^(2))/(e^(2)))` then `(d^(2)y)/(dx^(2))` equal to: a) `-(1)/(x)` b) `-(1)/(x^(2))` c) `(2)/(x^(2))` d) `-(2)/(x^(2))`

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 \( y = \log\left(\frac{x^2}{e^2}\right) \).

### Step 1: Simplify the expression for \( y \)

Using the properties of logarithms, we can rewrite the function:

\[ y = \log\left(\frac{x^2}{e^2}\right) = \log(x^2) - \log(e^2) \]

Using the property \( \log(a^b) = b \log(a) \):

\[ y = 2 \log(x) - 2 \log(e) \]

Since \( \log(e) = 1 \):

\[ y = 2 \log(x) - 2 \]

### Step 2: Find the first derivative \( \frac{dy}{dx} \)

Now we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}(\log(x)) - 0 \]

The derivative of \( \log(x) \) is \( \frac{1}{x} \):

\[ \frac{dy}{dx} = 2 \cdot \frac{1}{x} = \frac{2}{x} \]

### Step 3: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2}{x}\right) \]

Using the power rule, we can rewrite \( \frac{2}{x} \) as \( 2x^{-1} \):

\[ \frac{d^2y}{dx^2} = 2 \cdot \frac{d}{dx}(x^{-1}) = 2 \cdot (-1)x^{-2} = -\frac{2}{x^2} \]

### Final Answer

Thus, the second derivative \( \frac{d^2y}{dx^2} \) is:

\[ \frac{d^2y}{dx^2} = -\frac{2}{x^2} \]

The correct option is (d) \( -\frac{2}{x^2} \).

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