Question

If \( y = x^x \), then \( \frac{dy}{dx} \) is:

• A. \( x^x (\ln x + 1) \)
• B. \( x \times x^{x - 1} \)
• C. \( x \times (x - 1) \)
• D. \( x \ln x \)

Hint:

For functions like \( y = x^x \), take logarithms first: Use \( \ln y = x \ln x \) and then differentiate implicitly.

Answer 1

The Correct Option is A

Step 1: Rewrite the function using logarithms.

Given \( y = x^x \). Take natural log on both sides:
\[ \ln y = \ln(x^x) = x \ln x \quad \cdots (1) \]
Step 2: Differentiate both sides using implicit differentiation.

Differentiate equation (1) with respect to \( x \):
\[ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}(x \ln x) = \ln x + 1 \]
Step 3: Multiply both sides by \( y = x^x \).

\[ \frac{dy}{dx} = x^x (\ln x + 1) \]