Question

If \(y = log(\frac{x^5}{e^5})\), then \(\frac{d^2y}{dx^2}\) is,

• A. \(\frac{-5}{x^2}\)
• B. \(\frac{-20}{x^4}\)
• C. \(\frac{x^3}{e^2}\)
• D. \(\frac{-20x^3}{e^2}\)

Answer 1

The Correct Option is A

Given the function \(y = \log\left(\frac{x^5}{e^5}\right)\), we need to find \(\frac{d^2y}{dx^2}\).

First, use the properties of logarithms to simplify the expression: \[\log\left(\frac{x^5}{e^5}\right) = \log(x^5) - \log(e^5)\]

Since \(\log(e^5)=5\) (because \(\log(e)=1\)), the equation simplifies to: \[y = 5\log(x) - 5\]

Now, differentiate \(y\) with respect to \(x\): \[\frac{dy}{dx} = \frac{d}{dx}[5\log(x) - 5] = 5 \cdot \frac{1}{x} = \frac{5}{x}\]

To find the second derivative \(\frac{d^2y}{dx^2}\), differentiate \(\frac{dy}{dx}\) with respect to \(x\): \[\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{5}{x}\right) = 5 \cdot \frac{-1}{x^2} = \frac{-5}{x^2}\]

Thus, the second derivative is: \(\frac{d^2y}{dx^2} = \frac{-5}{x^2}\).

The correct answer is: \(\frac{-5}{x^2}\).