Answer is: option2

(B) \( x^x (1 + \ln x) \)

Solution:

Taking the natural logarithm on both sides:

\[ \ln y = \ln(x^x) \]

Using the logarithm property \( \ln(a^b) = b \ln a \):

\[ \ln y = x \ln x \]

Differentiating both sides implicitly with respect to \( x \):

\[ \frac{d}{dx} (\ln y) = \frac{d}{dx} (x \ln x) \] Using the derivative rule: \[ \frac{1}{y} \frac{dy}{dx} = 1 \cdot \ln x + x \cdot \frac{1}{x} \] \[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

Multiplying both sides by \( y = x^x \):

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

Looking at the answer choices, the correct one is:

\[ \boxed{x^x (1 + \ln x)} \]