Question

If \(y = (x - 1)\log_{e}(x - 1)\), then \(\frac{d^{2}y}{dx^{2}}\) at \(x = 3\) is

• A. e
• B. \(e^{2}\)
• C. 3
• D. \(\frac{1}{2}\)
• E. \(\frac{1}{4}\)

Hint:

\(\frac{d}{dx} \ln(g(x)) = \frac{g'(x)}{g(x)}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Use product rule and chain rule. Let \(u = x-1\). Then \(y = u \ln u\).

Step 2: Detailed Explanation:
\(\frac{dy}{dx} = \ln(x-1) + (x-1) \cdot \frac{1}{x-1} = \ln(x-1) + 1\)
\(\frac{d^2y}{dx^2} = \frac{1}{x-1}\)
At \(x = 3\): \(\frac{d^2y}{dx^2} = \frac{1}{3-1} = \frac{1}{2}\)

Step 3: Final Answer:
\(\frac{d^2y}{dx^2} = \frac{1}{2}\).