The derivative is  supremely useful because there are so many ways to think of it.  It is the instantaneous rate of change, so velocity and acceleration can be found when give a position function, it also describes sensitivity of how change in the input variable is reflected in the change of the output variable, and it has geometric meaning as the slope of the tangent to the graph of a function.  This chapter explores how all these understandings of the derivative lead to applications.

The derivative tells us a great deal about the shape of a curve.  The derivative gives us precise information about how the curve bends and exactly where it turns.  Rates of change will be deduced where we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point.  The key to recovering functions from derivatives is the Mean Value Theorem, the theorem whose corollaries provide the gateway to integral calculus, which begins in chapter 6.

This chapter begins where the early developers of calculus began, with the derivative as the slope of the tangent line and the insights this gives us into finding the greatest and least values of a function.

​(Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)