1. **State the problem:** Differentiate the function $$y(x) = \frac{x^3 + 1}{5 - 2x^3}$$ with respect to $x$. 2. **Recall the quotient rule:** For a function $$y = \frac{u(x)}{v(x)}$$, the derivative is $$y' = \frac{u'v - uv'}{v^2}$$. 3. **Identify $u$ and $v$:** $$u = x^3 + 1$$ $$v = 5 - 2x^3$$ 4. **Compute derivatives:** $$u' = 3x^2$$ $$v' = -6x^2$$ 5. **Apply the quotient rule:** $$y' = \frac{3x^2(5 - 2x^3) - (x^3 + 1)(-6x^2)}{(5 - 2x^3)^2}$$ 6. **Simplify numerator:** $$= \frac{3x^2 \cdot 5 - 3x^2 \cdot 2x^3 + 6x^2 \cdot x^3 + 6x^2}{(5 - 2x^3)^2}$$ $$= \frac{15x^2 - 6x^5 + 6x^5 + 6x^2}{(5 - 2x^3)^2}$$ 7. **Combine like terms:** $$= \frac{15x^2 + 6x^2}{(5 - 2x^3)^2} = \frac{21x^2}{(5 - 2x^3)^2}$$ **Final answer:** $$y'(x) = \frac{21x^2}{(5 - 2x^3)^2}$$