In the previous chapters, we explored the mathematical foundations of the derivative, delving into its properties and direct applications such as determining the slope of a curve at a point and calculating the instantaneous rate of change of specific variables. We also examined how derivatives allow us to understand the behavior of functions, analyze motion, and solve problems involving velocity and acceleration.
However, the true power of derivatives extends far beyond these fundamental concepts. They are not just tools for theoretical exploration but are indispensable in solving practical problems across various fields such as physics, engineering, economics, and biology. In this chapter, we will look into some of the most significant applications of the derivative: optimization and related rates.