1. **Problem 1: Find \( \frac{dy}{dx} \) given the implicit equation \( x^2 + xy + y^3 = 0 \).** 2. We use implicit differentiation. Differentiate both sides with respect to \( x \): $$\frac{d}{dx}(x^2) + \frac{d}{dx}(xy) + \frac{d}{dx}(y^3) = \frac{d}{dx}(0)$$ 3. Apply the rules: - \( \frac{d}{dx}(x^2) = 2x \) - For \( xy \), use product rule: \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \) - For \( y^3 \), use chain rule: \( 3y^2 \frac{dy}{dx} \) 4. Substitute: $$2x + x \frac{dy}{dx} + y + 3y^2 \frac{dy}{dx} = 0$$ 5. Group terms with \( \frac{dy}{dx} \): $$x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = -2x - y$$ 6. Factor out \( \frac{dy}{dx} \): $$\frac{dy}{dx}(x + 3y^2) = -2x - y$$ 7. Solve for \( \frac{dy}{dx} \): $$\frac{dy}{dx} = -\frac{2x + y}{x + 3y^2}$$ --- 8. **Problem 2: Given \( f'(x) = e^x + \cos x \) and \( f(0) = 0 \), find \( f(x) \).** 9. To find \( f(x) \), integrate \( f'(x) \): $$f(x) = \int (e^x + \cos x) dx = \int e^x dx + \int \cos x dx$$ 10. Integrate each term: $$\int e^x dx = e^x + C_1$$ $$\int \cos x dx = \sin x + C_2$$ 11. Combine constants into one \( C \): $$f(x) = e^x + \sin x + C$$ 12. Use initial condition \( f(0) = 0 \): $$0 = e^0 + \sin 0 + C = 1 + 0 + C \implies C = -1$$ 13. Final function: $$f(x) = e^x + \sin x - 1$$