Chapter 2
Models of Growth: Rates of Change
2.2 The Derivative: Instantaneous Rate of Change
2.2.4 The Derivative
To calculate instantaneous rate of change from average rate of change, we have to let the change in the independent variable approach zero without actually letting it equal zero. We abbreviate the process of letting approach zero by “.” Now we give a mathematical name to the limiting value of average rates of change.
| Definition The limiting value of the difference quotient
as is called the derivative of \(s\) with respect to \(t\) (at the particular value of \(t\) in question) and is denoted by
.
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Of course, when the change in the independent variable approaches \(0\), the change in the dependent variable usually does also. Thus, \({ds} / {dt}\) is what \({\Delta s} / {\Delta t}\) approaches when \(\Delta t\) and \(\Delta s\) both approach zero. Think back to the zooming process earlier in this section. When we look at a segment of the graph of \(s\) (as a function of \(t\)) that is so small it appears straight, then \(ds / dt\) is the slope of that straight line.
We now have a notation,
to represent the instantaneous rate of change. In the transition from difference quotient to derivative, each upper case delta, standing for difference, is replaced by a lower case d, which stands for differential.
The calculation in Example 1 shows that, if
then
The process of calculating a derivative is called differentiation, and the verb form is differentiate. Thus, in Example 1 the instruction might be "Differentiate \(s=ct^2\) with respect to \(t\)" — or just "Differentiate \(s=ct^2\)" if it is clear from the context that \(c\) is a constant and \(t\) is the independent variable. In this and following chapters, we will develop formulas for calculating derivatives, and you will often see instructions of the form "Differentiate the following function."
