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 $\Delta t$ approach zero without actually letting it equal zero. We abbreviate the process of letting $\Delta t$ approach zero by “$\Delta t\to 0$.” Now we give a mathematical name to the limiting value of average rates of change.

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