Given a Cobb-Douglas production function, we can find the marginal product of capital and labor by taking partial derivatives. A general Cobb-Douglas production function is typically represented as: $$Q = A K^\alpha L^\beta$$ where: $Q$ is the total output $A$ is the total factor productivity $K$ is the capital input $L$ is the labor input $\alpha$ is the output elasticity of capital $\beta$ is the output elasticity of labor 1. Marginal Product of Capital (MPK): The marginal product of capital is the additional output produced by employing one more unit of capital, holding labor constant. It is found by taking the partial derivative of the production function with respect to capital ($K$). $$MPK = \frac{\partial Q}{\partial K}$$ $$MPK = \frac{\partial}{\partial K} (A K^\alpha L^\beta)$$ Treating $A$ and $L^\beta$ as constants: $$MPK = A L^\beta \frac{\partial}{\partial K} (K^\alpha)$$ $$MPK = A L^\beta (\alpha K^{\alpha-1})$$ $$MPK = \boxed{\alpha A K^{\alpha-1} L^\beta}$$ 2. Marginal Product of Labor (MPL): The marginal product of labor is the additional output produced by employing one more unit of labor, holding capital constant. It is found by taking the partial derivative of the production function with respect to labor ($L$). $$MPL = \frac{\partial Q}{\partial L}$$ $$MPL = \frac{\partial}{\partial L} (A K^\alpha L^\beta)$$ Treating $A$ and $K^\alpha$ as constants: $$MPL = A K^\alpha \frac{\partial}{\partial L} (L^\beta)$$ $$MPL = A K^\alpha (\beta L^{\beta-1})$$ $$MPL = \boxed{\beta A K^\alpha L^{\beta-1}}$$