Optimization with derivatives plays a crucial role in various economic scenarios, helping maximize profit or minimize cost. Let's consider an example: a company manufactures and sells a product. The profit derived from selling each unit of the product can be represented by a function, let's say P(x). To maximize profit, we need to find the value of x that yields the highest P(x).

Using derivatives to optimize, we first find the derivative of P(x) with respect to x, denoted as P'(x). We then set P'(x) equal to zero and solve for x to find the critical points. By checking the second derivative at these points, we can determine whether they correspond to maximum or minimum values.

For instance, let's say P(x) = -2x^2 + 100x - 500. Taking the derivative, we get P'(x) = -4x + 100. Setting P'(x) equal to zero, we find x = 25. Calculating the second derivative, we get P''(x) = -4. Since the second derivative is negative, the critical point x = 25 corresponds to a maximum profit.

Optimization with derivatives enables economists to make data-driven decisions and find the most profitable outcomes. By understanding the underlying mathematics, we can analyze and optimize various economic situations, ultimately leading to greater financial success and productivity.