1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{x^4}{2} + x^3.$$\n\n2. **Recall the derivative rules:**\n- The derivative of $x^n$ is $nx^{n-1}$.\n- The derivative of a sum is the sum of the derivatives.\n- Constants can be factored out of the derivative.\n\n3. **Apply the derivative:**\n$$\frac{d}{dx}\left(\frac{x^4}{2} + x^3\right) = \frac{d}{dx}\left(\frac{1}{2}x^4\right) + \frac{d}{dx}(x^3).$$\n\n4. **Differentiate each term:**\n- For $\frac{1}{2}x^4$, derivative is $\frac{1}{2} \cdot 4x^{3} = 2x^{3}$.\n- For $x^3$, derivative is $3x^{2}$.\n\n5. **Combine the results:**\n$$f'(x) = 2x^{3} + 3x^{2}.$$\n\n**Final answer:** $$\boxed{2x^{3} + 3x^{2}}.$$