Question

If \(y=log[\frac{x^2}{e^2}]\) then value of \(\frac{d^2y}{dx^2}\) is:

• A. \(\frac{x^2}{e^4}\)
• B. \(\frac{-2}{x^2}\)
• C. \(\frac{e}{x^2}\)
• D. 2x+log2

Answer 1

The Correct Option is B

To find the second derivative of the function \( y=\log\left(\frac{x^2}{e^2}\right) \) with respect to \( x \), we follow these steps:

1. Using the properties of logarithms, rewrite the function:
   \( y=\log(x^2)-\log(e^2) \).
   Since \( \log(e^2)=2 \), this simplifies to:
   \( y=\log(x^2)-2 \).
2. Apply the property of logarithms \( \log(a^b)=b\log(a) \):
   \( y=2\log(x)-2 \).
3. Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \):
   \( \frac{dy}{dx} = 2\cdot\frac{1}{x} = \frac{2}{x} \).
4. Differentiate again to find \( \frac{d^2y}{dx^2} \):
   \( \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2}{x}\right) = 2\cdot\left(-\frac{1}{x^2}\right)=\frac{-2}{x^2} \).

Thus, the value of \( \frac{d^2y}{dx^2} \) is \( \frac{-2}{x^2} \).