Optimization problems can become more challenging when there are constraints involved. In constrained optimization, we aim to find the maximum or minimum value of a function while satisfying certain conditions or limitations. These conditions can be represented as equations or inequalities.

To solve constrained optimization problems using derivatives, we often utilize the method of Lagrange multipliers. This technique allows us to incorporate the constraints into the objective function by introducing additional variables and Lagrange multipliers. By taking the partial derivatives of the modified objective function with respect to both the original variables and the Lagrange multipliers, we can find critical points that satisfy the constraints.

Let's consider an example. Suppose we want to maximize the function f(x, y) = 2x + 3y subject to the constraint x^2 + y^2 = 25. To solve this problem, we introduce a Lagrange multiplier λ and form the equation F(x, y, λ) = 2x + 3y - λ(x^2 + y^2 - 25). By taking the partial derivatives of F with respect to x, y, and λ, and setting them equal to zero, we can find the critical points which satisfy the constraint. We then evaluate the function at these critical points to determine the maximum or minimum value.

By understanding constrained optimization and utilizing techniques like Lagrange multipliers, we can solve complex problems that have real-world applications in various fields, such as engineering, finance, and operations research.