In Chapter 2, we learned how to find the slope of a tangent to a curve as the limit of the slopes of secant lines.  In section 2.4, a formula was derived for the slope of the tangent at an arbitrary point (a, 1/a) on the graph of the function f(x) = 1/x and showed it was -1/a^2.
This result is more powerful than it might appear at first glance, as it gives us a simple way to calculate the instantaneous rate of change of f at any point.  The study of rates of change of functions is called differential calculus, and the formula -1/a^2 was our first look at a derivative.  The derivative was the 17th-century breakthrough that enabled mathematicians to unlock the secrets of planetary motion and gravitational attraction -- of objects changing position over time.  We will learn many uses for derivatives in Chapter 5, but first, in the next two chapters, we will focus on what derivatives are and how they work.
​ (Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)