1. **State the problem:** We are given the function $$y = \frac{x^6}{2} + \frac{x^4}{4}$$ and asked to find its derivative $$\frac{dy}{dx}$$ and simplify the answer. 2. **Recall the derivative rules:** The derivative of $$x^n$$ with respect to $$x$$ is $$nx^{n-1}$$. Also, the derivative of a sum is the sum of the derivatives. 3. **Apply the derivative to each term:** $$\frac{d}{dx}\left(\frac{x^6}{2}\right) = \frac{1}{2} \cdot \frac{d}{dx}(x^6) = \frac{1}{2} \cdot 6x^{5} = 3x^{5}$$ $$\frac{d}{dx}\left(\frac{x^4}{4}\right) = \frac{1}{4} \cdot \frac{d}{dx}(x^4) = \frac{1}{4} \cdot 4x^{3} = x^{3}$$ 4. **Sum the derivatives:** $$\frac{dy}{dx} = 3x^{5} + x^{3}$$ 5. **Simplify the expression:** Factor out the common term $$x^{3}$$: $$\frac{dy}{dx} = x^{3}(3x^{2} + 1)$$ **Final answer:** $$\frac{dy}{dx} = x^{3}(3x^{2} + 1)$$