1. **Problem:** Find the derivative of the implicit function given by $$x^3 + y^2 = -x^5$$. 2. **Formula and rules:** Use implicit differentiation. Differentiate both sides with respect to $x$, remembering that $y$ is a function of $x$, so apply the chain rule: $$\frac{d}{dx}[y^2] = 2y y'$$. 3. **Differentiate:** $$\frac{d}{dx}[x^3] + \frac{d}{dx}[y^2] = \frac{d}{dx}[-x^5]$$ $$3x^2 + 2y y' = -5x^4$$ 4. **Solve for $y'$:** $$2y y' = -5x^4 - 3x^2$$ $$y' = \frac{-5x^4 - 3x^2}{2y}$$ 5. **Simplify:** $$y' = \frac{-5x^4 - 3x^2}{2y}$$ 6. **Boxed final answer:** $$\boxed{y' = \frac{-5x^4 - 3x^2}{2y}}$$ --- **Note:** Your original answer missed the $-3x^2$ term in the numerator and had a different denominator. The correct derivative includes both terms in the numerator and $2y$ in the denominator. q_count: 1