Rates of change of functions

The rate of change of a function refers to the amount the function's output increases or decreases for each unit of change in the input.

For example, if a vehicle is traveling at 55 miles per hour, the distance function is increasing 55 miles for each hour of travel. So the rate of change of the distance function is a constant 55 mi/hr.

More typically, a vehicle's speed will change. Accordingly, the rate of change of the distance function will have different values at different times. The rate of change might be 30 mi/hr at one instant and 60 mi/hr at another instant.

Even if a vehicle doesn't travel at 30 mi/hr for a full hour, the rate of change at one instant can be 30 mi/hr. That means that if the vehicle were to continue at that speed for an hour, it would travel 30 miles. In general, if a function's rate of change at one instant is r output units per input unit, and if the function continues to change for an entire unit at that rate, then its total change will be r output units.

A function with a constant rate of change of 0 is not changing. If the function's rate of change is a positive constant, then the function is "steadily" increasing. If the rate of change is a negative constant, then the function is decreasing similarly. Such functions are called linear functions because their graphs are straight lines. The rate of change gives the slope of the line.

For many common functions, the rate of change is a constant times the (changing) value of the function. For example, an interest-bearing account periodically is increased by a certain percentage of the amount in the account. As the value of the account grows, the amount of interest grows, even though the percentage stays the same. Or a population of animals may die out, again with a certain portion of the current population dying each year. Such functions are called exponential functions because their explicit function expressions involve exponents.

The rates of change of other functions also show patterns. In fact, from a function f(x) you can make a function f '(x) with the same input values but whose output values are the rates of change. Such a function is called the derived function, or derivative, of f(x). The value of f '(x) gives the slope of a line tangent to the graph of f(x) at that point in terms of the output value for some other input value.