1. **State the problem:** We want to find the derivative of the function $f(x) = a\sqrt{x}$, where $a$ is a constant. 2. **Rewrite the function:** Recall that $\sqrt{x} = x^{1/2}$. So the function becomes $$f(x) = a x^{1/2}$$ 3. **Apply the power rule:** The derivative of $x^{n}$ with respect to $x$ is $nx^{n-1}$. Using this rule, $$f'(x) = a \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{a}{2} x^{-\frac{1}{2}}$$ 4. **Simplify the result:** We can rewrite $x^{-1/2}$ as $\frac{1}{\sqrt{x}}$: $$f'(x) = \frac{a}{2 \sqrt{x}}$$ Thus, the derivative of $a\sqrt{x}$ is $\frac{a}{2\sqrt{x}}$.