1. **Problem 1:** Given the equation $$x^2 y = \sin y = 3x$$, find $$\frac{dy}{dx}$$ using implicit differentiation. Step 1: Clarify the equation. It seems there is a typo or confusion. Assuming the equation is $$x^2 y + \sin y = 3x$$. Step 2: Differentiate both sides with respect to $$x$$. Using product rule on $$x^2 y$$: $$\frac{d}{dx}(x^2 y) = 2x y + x^2 \frac{dy}{dx}$$. Using chain rule on $$\sin y$$: $$\frac{d}{dx}(\sin y) = \cos y \frac{dy}{dx}$$. Right side derivative: $$\frac{d}{dx}(3x) = 3$$. Step 3: Write the differentiated equation: $$2x y + x^2 \frac{dy}{dx} + \cos y \frac{dy}{dx} = 3$$ Step 4: Group terms with $$\frac{dy}{dx}$$: $$x^2 \frac{dy}{dx} + \cos y \frac{dy}{dx} = 3 - 2x y$$ Step 5: Factor $$\frac{dy}{dx}$$: $$\frac{dy}{dx}(x^2 + \cos y) = 3 - 2x y$$ Step 6: Solve for $$\frac{dy}{dx}$$: $$\frac{dy}{dx} = \frac{3 - 2x y}{x^2 + \cos y}$$ --- 2. (a) **Problem 2(a):** Differentiate $$y = 4^x \cdot \cos(3x)$$. Step 1: Use product rule: $$\frac{d}{dx}(uv) = u'v + uv'$$. Let $$u = 4^x$$ and $$v = \cos(3x)$$. Step 2: Differentiate $$u$$: $$u' = 4^x \ln 4$$. Step 3: Differentiate $$v$$ using chain rule: $$v' = -\sin(3x) \cdot 3 = -3 \sin(3x)$$. Step 4: Apply product rule: $$\frac{dy}{dx} = 4^x \ln 4 \cdot \cos(3x) + 4^x \cdot (-3 \sin(3x))$$ Step 5: Simplify: $$\frac{dy}{dx} = 4^x \left( \ln 4 \cos(3x) - 3 \sin(3x) \right)$$ --- 2. (b) **Problem 2(b):** Find the third derivative of $$y = 5x^2 e^x$$. Step 1: First derivative using product rule: $$y' = \frac{d}{dx}(5x^2) e^x + 5x^2 \frac{d}{dx}(e^x) = 10x e^x + 5x^2 e^x = e^x (10x + 5x^2)$$ Step 2: Second derivative: $$y'' = \frac{d}{dx} \left( e^x (10x + 5x^2) \right) = e^x (10x + 5x^2) + e^x (10 + 10x) = e^x (10x + 5x^2 + 10 + 10x) = e^x (5x^2 + 20x + 10)$$ Step 3: Third derivative: $$y''' = \frac{d}{dx} \left( e^x (5x^2 + 20x + 10) \right) = e^x (5x^2 + 20x + 10) + e^x (10x + 20) = e^x (5x^2 + 20x + 10 + 10x + 20) = e^x (5x^2 + 30x + 30)$$ --- **Final answers:** 1. $$\frac{dy}{dx} = \frac{3 - 2x y}{x^2 + \cos y}$$ 2. (a) $$\frac{dy}{dx} = 4^x \left( \ln 4 \cos(3x) - 3 \sin(3x) \right)$$ 2. (b) $$\frac{d^3 y}{dx^3} = e^x (5x^2 + 30x + 30)$$