Derivative

The derivative is often called the "instantaneous" rate of change. A rate of change is simply a comparison of the change in one quantity to the simultaneous change in a second quantity. For instance, the amount of money your employer owes you compared to the length of time you worked for him determines your rate of pay. The comparison is made in the form of a ratio, dividing the change in the first quantity by the change in the second quantity. When both changes occur during an infinitely short period of time (in the same instant), the rate is said to be "instantaneous," and then the ratio is called the derivative.

To better understand what is meant by an instantaneous rate of change, consider the graph of a straight line (see Figure 1).

The line's slope is defined to be the ratio of the rise (vertical change between any two points) to the run (simultaneous horizontal change between the same two points). This means that the slope of a straight line is a rate, specifically, the line's rate of rise with respect to the horizontal axis. It is the simplest type of rate because it is constant, the same between any two points, even two points that are arbitrarily close together. Roughly speaking, arbitrarily close together means you can make them closer than any positive amount of separation. The derivative of a straight line, then, is the same for every point on the line and is equal to the slope of the line.

Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Determining the derivative of a curve is somewhat more difficult, because its instantaneous rate of rise changes from point to point (see Figure 2).

We can estimate a curve's rate of rise at any particular point, though, by noticing that any section of a curve can be approximated by replacing it with a straight line. Since we know how to determine the slope of a straight line, we can approximate a curve's rate of rise at any point, by determining the slope of an approximating line segment. The shorter the approximating line segment becomes, the more accurate the estimate becomes. As the length of the approximating line segment becomes arbitrarily short, so does its rise and its run. Just as in the case of the straight line, an arbitrarily short rise and run can be shorter than any given positive pair of distances. Thus, their ratio is the instantaneous rate of rise of the curve at the point or the derivative. In this case the derivative is different at every point, and equal to the slope of the tangent at each point. (A tangent is a straight line that intersects a curve at a single point.)