Given a Cobb-Douglas production function, we can find the marginal product of capital and labor by taking partial derivatives.

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:

$$MPK = A L^\beta \frac{\partial}{\partial K} (K^\alpha)$$ $$MPK = A L^\beta (\alpha K^{\alpha-1})$$ $$MPK = \boxed{\alpha A K^{\alpha-1} L^\beta}$$

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:

$$MPL = A K^\alpha \frac{\partial}{\partial L} (L^\beta)$$ $$MPL = A K^\alpha (\beta L^{\beta-1})$$ $$MPL = \boxed{\beta A K^\alpha L^{\beta-1}}$$