The main tool that you’ll use in differential calculus is called the derivative. All of the problems that
you’ll encounter in differential calculus make use of the derivative, so your goal should be to become an
expert at finding, or “taking,” derivatives by the end of Chapter 6. However, before you learn a simple
way to take a derivative, your teacher will probably make you learn how derivatives are calculated by
teaching you something called the “Definition of the Derivative.”
DERIVING THE FORMULA
The best way to understand the definition of the derivative is to start by looking at the simplest continuous
function: a line. As you should recall, you can determine the slope of a line by taking two points on that
line and plugging them into the slope formula.
m = m stands for slope.For example, suppose a line goes through the points (3, 7) and (8, 22). First, you subtract the y-
coordinates (22 − 7) = 15. Next, subtract the corresponding x-coordinates (8 − 3) = 5. Finally, divide the
first number by the second: = 3. The result is the slope of the line: m = 3.
Notice that you can use the coordinates in reverse order and
still get the same result. It doesn’t matter in which order you do
the subtraction as long as you’re consistent.Let’s look at the graph of that line. The slope measures the steepness of the line, which looks like the
following: