Summary: Chapter 3
Introduction to Derivatives
In this chapter we will be understand what a derivative is, how to tell if it exists or not, and how to find it if it
does. The ideas of limits and continuity of a function are pillars on which we build the derivative so we begin
with those concepts.
Calculus required continuity, and continuity was supposed to require the
infinitely little; but nobody could discover what the infinitely little might
Yet in another way, calculus is fundamentally naive, almost childish in
its optimism. Experience teaches us that change can be sudden, dis-
continuous, and wrenching. Calculus draws its power by refusing to see
that. It insists on a world without accidents, where one thing leads log-
ically to another. Give me the initial conditions and the law of motion,
and with calculus I can predict the future or better yet, reconstruct the
past. I wish I could do that now.
Calculus has its limits.
3.1 Limits and Continuity