1. **State the problem:** We need to find the derivative with respect to $x$ of the function $$f(x) = \ln(\sqrt{x}).$$ 2. **Rewrite the function:** Recall that $\sqrt{x} = x^{\frac{1}{2}}$, so $$f(x) = \ln(x^{\frac{1}{2}}).$$ 3. **Use logarithm properties:** Using the property $\ln(a^b) = b \ln(a)$, we get $$f(x) = \frac{1}{2} \ln(x).$$ 4. **Differentiate:** The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$, so $$\frac{d}{dx} f(x) = \frac{d}{dx} \left( \frac{1}{2} \ln(x) \right) = \frac{1}{2} \cdot \frac{1}{x} = \frac{1}{2x}.$$