1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = \frac{5x^7 - 3x^9}{2x}$$ and simplify the result. 2. **Rewrite the function:** Simplify the expression by dividing each term in the numerator by $2x$: $$y = \frac{5x^7}{2x} - \frac{3x^9}{2x}$$ 3. **Simplify powers of $x$:** $$y = \frac{5}{2} x^{7-1} - \frac{3}{2} x^{9-1} = \frac{5}{2} x^6 - \frac{3}{2} x^8$$ 4. **Apply the power rule for derivatives:** The power rule states that $$\frac{d}{dx} x^n = n x^{n-1}$$. 5. **Differentiate each term:** $$\frac{dy}{dx} = \frac{5}{2} \cdot 6 x^{6-1} - \frac{3}{2} \cdot 8 x^{8-1} = 15 x^5 - 12 x^7$$ 6. **Final answer:** $$\boxed{\frac{dy}{dx} = 15 x^5 - 12 x^7}$$ This is the simplest form of the derivative.