Answer is: option2

\( \frac{x^{\ln \sqrt{x}} \ln x}{x} \)

Solution:

We are given:

\[ y = x^{\ln \sqrt{x}} \]

Since \( \ln \sqrt{x} = \ln x^{1/2} = \frac{1}{2} \ln x \), we rewrite:

\[ y = x^{\frac{1}{2} \ln x} \]

Taking the natural logarithm on both sides:

\[ \ln y = \frac{1}{2} \ln x \cdot \ln x = \frac{1}{2} (\ln x)^2 \] Differentiating both sides implicitly: \[ \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \cdot 2 \ln x \cdot \frac{1}{x} = \frac{\ln x}{x} \]

Multiplying both sides by \( y = x^{\frac{1}{2} \ln x} \):

\[ y' = x^{\frac{1}{2} \ln x} \cdot \frac{\ln x}{x} \]

Rewriting \( x^{\frac{1}{2} \ln x} \) as \( x^{\ln \sqrt{x}} \), we get:

\[ y' = \frac{x^{\ln \sqrt{x}} \ln x}{x} \]

which matches option (B):

\[ \boxed{\frac{x^{\ln \sqrt{x}} \ln x}{x}} \]