1. The problem is to find the derivative of the function $f(x) = x\sqrt{x}$. 2. First, rewrite the function using exponent notation: $f(x) = x \cdot x^{1/2} = x^{3/2}$. 3. Now, use the power rule for derivatives, which states that $\frac{d}{dx}x^n = nx^{n-1}$. 4. Apply the power rule to $f(x) = x^{3/2}$: $$f'(x) = \frac{3}{2}x^{\frac{3}{2} - 1} = \frac{3}{2}x^{\frac{1}{2}}.$$ 5. Rewrite the result in terms of square roots if preferred: $$f'(x) = \frac{3}{2} \sqrt{x}.$$ Final answer: The derivative of $x\sqrt{x}$ is $\frac{3}{2}\sqrt{x}$.