1. **Problem Statement:** Given the equation $$x^2 y = \sin y = 3x$$, find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Step 1: Understand the equation** The equation as given is ambiguous because it uses two equal signs. Assuming the problem means $$x^2 y + \sin y = 3x$$. 3. **Step 2: Differentiate both sides with respect to $$x$$** Using implicit differentiation: - Differentiate $$x^2 y$$ using the product rule: $$\frac{d}{dx}(x^2 y) = 2x y + x^2 \frac{dy}{dx}$$ - Differentiate $$\sin y$$: $$\cos y \frac{dy}{dx}$$ - Differentiate $$3x$$: $$3$$ 4. **Step 3: Write the differentiated equation** $$2x y + x^2 \frac{dy}{dx} + \cos y \frac{dy}{dx} = 3$$ 5. **Step 4: Group terms with $$\frac{dy}{dx}$$** $$x^2 \frac{dy}{dx} + \cos y \frac{dy}{dx} = 3 - 2x y$$ 6. **Step 5: Factor out $$\frac{dy}{dx}$$** $$\frac{dy}{dx} (x^2 + \cos y) = 3 - 2x y$$ 7. **Step 6: Solve for $$\frac{dy}{dx}$$** $$\frac{dy}{dx} = \frac{3 - 2x y}{x^2 + \cos y}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{3 - 2x y}{x^2 + \cos y}}$$