1. **State the problem:** Find the derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $$5y^2 = 2x^3 - 5y$$ 2. **Recall the formula and rules:** - Use implicit differentiation: differentiate both sides with respect to $x$. - Remember $\frac{d}{dx}[y] = \frac{dy}{dx}$ since $y$ is a function of $x$. - Use the chain rule for $y^2$ and $y$ terms. 3. **Differentiate both sides:** $$\frac{d}{dx}[5y^2] = \frac{d}{dx}[2x^3 - 5y]$$ 4. **Apply differentiation:** - Left side: $5 \cdot 2y \frac{dy}{dx} = 10y \frac{dy}{dx}$ - Right side: $6x^2 - 5 \frac{dy}{dx}$ So we have: $$10y \frac{dy}{dx} = 6x^2 - 5 \frac{dy}{dx}$$ 5. **Group $\frac{dy}{dx}$ terms:** $$10y \frac{dy}{dx} + 5 \frac{dy}{dx} = 6x^2$$ $$\frac{dy}{dx}(10y + 5) = 6x^2$$ 6. **Solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = \frac{6x^2}{10y + 5} = \frac{6x^2}{10y + 5}$$ 7. **Simplify denominator if desired:** $$\frac{dy}{dx} = \frac{6x^2}{10y + 5}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{6x^2}{10y + 5}}$$