Page 75 - Calculus Workbook For Dummies

Chapter 5

Getting the Big Picture:

Differentiation Basics

In This Chapter

The ups and downs of finding slope and rate
The difference quotient: the other DQ

ifferentiation is the process of finding derivatives. The derivative is one of the most
Dimportant inventions in the history of mathematics and one of mathematics’ most
powerful tools. I’m sure you will feel both a deep privilege as you do the practice problems
below — and a keen sense of indebtedness to the great mathematicians of the past. Yeah,
yeah, yeah.

The Derivative: A Fancy Calculus

Word for Slope and Rate

A derivative of a function tells you how fast the output variable (like y) is changing compared
to the input variable (like x). For example, if y is increasing 3 times as fast as x — like with the
line y = 3 x + 5 — then you say that the derivative of y with respect to x equals 3, and you
dy dy 3
write = 3. This, of course, is the same as = , and that means nothing more than
dx dx 1
3
saying that the rate of change of y compared to x is 3-to-1, or that the line has a slope of .
1
The following problems emphasize the fact that a derivative is basically just a rate or a slope.
So to solve these problems, all you have to do is answer the questions as if they had asked
you to determine a rate or a slope instead of a derivative.

Q. What’s the derivative of y = 4x – 5?
You can think of the derivative dy
A. The answer is 4. You know, of course, that as basically rise . dx
the slope of y = 4x – 5 is 4, right? No? Egad! run
Any line of the form y = mx + b has a slope
equal to m. I hope that rings a bell. The
derivative of a line or curve is the same
thing as its slope, so the derivative of this
line is 4.