Question

If \(y = logx^x\), then the value of \((\frac{dy}{dx})^2\) at \(x = e^2\) is

• A. \(4\)
• B. \(e\)
• C. \(e^2\)
• D. \(9\)

Hint:

Rewrite log(x^x) as x log x and differentiate.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The logarithm here is natural. By the power rule of logs, \(\log x^x=x\log x\).

Step 2: Differentiate:
\[ \frac{dy}{dx}=\log x+x\cdot\frac1x=\log x+1 \]

Step 3: Evaluate at x = e squared:
\(\log e^2=2\), so \(\dfrac{dy}{dx}=2+1=3\).

Step 4: Square:
\(\left(\dfrac{dy}{dx}\right)^2=9\), option (D).

Final Answer:
dy/dx = log x + 1 equals 3 at x = e^2, so its square is 9. \[ \boxed{9} \]