Units and Differentiation

Differentiation is a sophisticated form of division. It is defined as the limit

and often denoted

If, for example, the function f(t) has values measured in feet and the variable t has values measured in seconds then the derivative f'(t) is a limit of quantities measured in feet per second and is itself measured in feet per second.

The TI-92 screens below illustrate how a computer algebra system can keep track of the units in this kind of situation.

• The first line of this screen defines a constant g representing the acceleration of a free-falling object near the earth's surface. It is measured in units of meters per second^2.

• The second line defines a function h(t) that represents the height of an object at time t measured in seconds. The object is dropped from an initial height of 100 meters at time t = 0. Notice that when this function is computed at a time t measured in seconds the result is measured in meters.

• The third line illustrates this.

• The first line of this screen computes the derivative of the function h(t). The function v(t) is defined to be this derivative.

• The second line displays the derivative. Notice that when the value of the derivative is computed at a time t measured in seconds the result is measured in units of meters per second.

• The third line illustrates this.