1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{x}{2} + x.$$ 2. **Rewrite the function:** $$f(x) = \frac{x}{2} + x = \frac{x}{2} + \frac{2x}{2} = \frac{3x}{2}.$$ 3. **Recall the derivative rule:** The derivative of $cx$ where $c$ is a constant is $c$. 4. **Calculate the derivative:** $$\frac{d}{dx} \left( \frac{3x}{2} \right) = \frac{3}{2}.$$ 5. **Final answer:** The derivative is $$\frac{3}{2}.$$