1. **Find the derivative dy/dx for each function:** (a) Given $$y = 3x^4 - x^3 + x^2 + 25x$$ Use power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ $$\frac{dy}{dx} = 12x^3 - 3x^2 + 2x + 25$$ (b) Given $$y = 6x^3 - \frac{x^4}{4} + \frac{x^3}{3} - \frac{1}{x^2}$$ Rewrite $$-\frac{1}{x^2} = -x^{-2}$$ Derivative: $$\frac{dy}{dx} = 18x^2 - x^3 + x^2 + 2x^{-3} = 18x^2 - x^3 + x^2 + 2x^{-3}$$ Simplify: $$\frac{dy}{dx} = (18x^2 + x^2) - x^3 + 2x^{-3} = 19x^2 - x^3 + 2x^{-3}$$ (c) Given $$y = (2x^2 + 3x - 1) \left( \frac{1}{x} - 3x + 2 \right)$$ Use product rule $$\frac{d}{dx} [u v] = u' v + u v'$$ Let $$u = 2x^2 + 3x - 1$$ and $$v = x^{-1} - 3x + 2$$ Then: $$u' = 4x + 3$$ $$v' = -x^{-2} - 3$$ Derivative: $$\frac{dy}{dx} = (4x + 3)(x^{-1} - 3x + 2) + (2x^2 + 3x -1)(-x^{-2} -3)$$ (d) Given $$y = \frac{4x^2 + 4}{x^2 + 3x + 2}$$ Use quotient rule $$\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f' g - f g'}{g^2}$$ Let $$f = 4x^2 + 4, f' = 8x$$ $$g = x^2 + 3x + 2, g' = 2x + 3$$ Derivative: $$\frac{dy}{dx} = \frac{8x (x^2 + 3x + 2) - (4x^2 + 4)(2x + 3)}{(x^2 + 3x + 2)^2}$$ 2. **Implicit differentiation:** (a) $$7x^2 + 2xy^2 + 9y^4 = 0$$ Differentiate both sides w.r.t $$x$$: $$14x + 2y^2 + 4xy \frac{dy}{dx} + 36y^3 \frac{dy}{dx} = 0$$ Group $$\frac{dy}{dx}$$ terms: $$\frac{dy}{dx} (4xy + 36y^3) = -14x - 2y^2$$ Solve: $$\frac{dy}{dx} = \frac{-14x - 2y^2}{4xy + 36y^3}$$ (b) $$3x^2 y^3 + x z^2 y^2 + y^3 z x^4 + y^2 z = 0$$ Partially differentiate w.r.t $$x$$ using implicit rules including $$\frac{dy}{dx}$$ and $$\frac{dz}{dx}$$ (requires more variables; typically partial derivatives). Since the question is for $$dy/dx$$ implicit differentiation, focus on expressing derivatives in terms of $$dy/dx$$ and $$dz/dx$$ but this is a complex multi-variable implicit differentiation. 3. **Total differentiation:** (a) $$z = x^4 + 8xy + 3y^3$$ $$dz = 4x^3 dx + 8y dx + 8x dy + 9y^2 dy$$ Combine terms: $$dz = (4x^3 + 8y) dx + (8x + 9y^2) dy$$ (b) $$z = \frac{x - y}{x+1}$$ Rewrite: $$z = (x - y)(x + 1)^{-1}$$ Differentiate: $$dz = \left(1 \cdot (x+1)^{-1} + (x - y)(-1)(x+1)^{-2} \right) dx - (x+1)^{-1} dy$$ Simplify: $$dz = \left( \frac{1}{x+1} - \frac{x - y}{(x+1)^2} \right) dx - \frac{1}{x+1} dy$$ 4. **Marginal Analysis** (1)(a) Demand: $$P = 250 - 2Q$$ Total Revenue $$TR = P \times Q = (250 - 2Q) Q = 250Q - 2Q^2$$ (1)(b) Marginal Revenue (MR) is derivative of $$TR$$ w.r.t $$Q$$: $$MR = \frac{d}{dQ} (250Q - 2Q^2) = 250 - 4Q$$ (2)(a) $$TC = 4q^3 + 2q^2 - 25q$$ Marginal Cost $$MC = \frac{dTC}{dq} = 12q^2 + 4q - 25$$ Average Cost $$AC = \frac{TC}{q} = 4q^2 + 2q - 25$$ (2)(b) $$TC = (q^3 - 3q)(16 + 5q)$$ First expand: $$TC = 16q^3 - 48q + 5q^4 - 15 q^2 = 5q^4 + 16q^3 - 15 q^2 - 48 q$$ Marginal Cost: $$MC = \frac{dTC}{dq} = 20 q^3 + 48 q^2 - 30 q - 48$$ Average Cost: $$AC = \frac{TC}{q} = 5 q^3 + 16 q^2 - 15 q - 48$$ (3) $$TC = 100 + 2 Q + \frac{1}{10} Q^2$$ Average Cost: $$AC = \frac{TC}{Q} = \frac{100}{Q} + 2 + \frac{Q}{10}$$ Minimize $$AC$$: Derivate w.r.t $$Q$$ and set to zero: $$\frac{d AC}{d Q} = -\frac{100}{Q^2} + \frac{1}{10} = 0$$ Multiply both sides by $$Q^2$$: $$-100 + \frac{Q^2}{10} = 0$$ So: $$\frac{Q^2}{10} = 100 \Rightarrow Q^2 = 1000 \Rightarrow Q = \sqrt{1000} = 10 \sqrt{10}$$ Average Cost at minimum output: $$AC = \frac{100}{10 \sqrt{10}} + 2 + \frac{10 \sqrt{10}}{10} = \frac{10}{\sqrt{10}} + 2 + \sqrt{10}$$ Simplify: $$\frac{10}{\sqrt{10}} = \sqrt{10}$$ So, $$AC = \sqrt{10} + 2 + \sqrt{10} = 2 + 2 \sqrt{10}$$ 5. **Partial derivatives:** (1)(a) $$z = x^2 + x y + 2 y^3 - 5 x - 4 y + 20$$ $$z_x = 2 x + y - 5$$ $$z_y = x + 6 y^2 - 4$$ $$z_{xx} = 2$$ $$z_{yy} = 12 y$$ $$z_{xy} = z_{yx} = 1$$ (1)(b) $$z = \frac{5 x}{6 x - 7 y}$$ Use quotient rule: $$z_x = \frac{(5)(6 x - 7 y) - 5 x (6)}{(6 x - 7 y)^2} = \frac{30 x - 35 y - 30 x}{(6 x -7 y)^2} = \frac{-35 y}{(6 x - 7 y)^2}$$ $$z_y = \frac{0 - 5 x (-7)}{(6 x - 7 y)^2} = \frac{35 x}{(6 x - 7 y)^2}$$ (2)(a) Cobb-Douglas $$Q = 10 K^{0.5} L^{0.5}$$ Marginal product of capital: $$MP_K = \frac{\partial Q}{\partial K} = 10 \times 0.5 K^{-0.5} L^{0.5} = 5 \frac{L^{0.5}}{K^{0.5}} = 5 \sqrt{\frac{L}{K}}$$ Marginal product of labour: $$MP_L = \frac{\partial Q}{\partial L} = 10 \times 0.5 K^{0.5} L^{-0.5} = 5 \frac{K^{0.5}}{L^{0.5}} = 5 \sqrt{\frac{K}{L}}$$ Marginal rate of substitution capital for labour: $$MRS_{K,L} = \frac{MP_K}{MP_L} = \frac{5 \sqrt{L/K}}{5 \sqrt{K/L}} = \frac{\sqrt{L/K}}{\sqrt{K/L}} = \frac{L/K}{K/L} = \frac{L^2}{K^2} = \left( \frac{L}{K} \right)^2$$ Final answer summarized accordingly.