1. **Problem:** Given the equation $$x^2 y = \sin y = 3x$$, find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Step 1:** Clarify the equation. The equation as given seems ambiguous with two equal signs. Assuming the intended equation is $$x^2 y + \sin y = 3x$$. 3. **Step 2:** Differentiate both sides with respect to $$x$$ implicitly. - Differentiate $$x^2 y$$ using the product rule: $$\frac{d}{dx}(x^2 y) = 2x y + x^2 \frac{dy}{dx}$$. - Differentiate $$\sin y$$ using the chain rule: $$\cos y \frac{dy}{dx}$$. - 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:** $$\boxed{\frac{dy}{dx} = \frac{3 - 2x y}{x^2 + \cos y}}$$