Average vs. Instantaneous Rate of Change: Secant Slopes and Derivatives

A rate of change measures how much an output changes compared with a change in the input. In calculus, the two central versions are average rate of change over an interval and instantaneous rate of change at a single input.

Average rate of change uses two endpoints

For a function $f$, the average rate of change from $x=a$ to $x=b$ is

$$\frac{f(b)-f(a)}{b-a}.$$

Geometrically, this is the slope of the secant line through the two points $(a,f(a))$ and $(b,f(b))$.

The units are “output units per input unit.” If distance is measured in kilometers and time in hours, the rate has units of kilometers per hour.

Instantaneous rate of change is a derivative

To describe the rate at exactly one input, shrink the interval around that input. At $x=a$,

$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h},$$

provided the limit exists.

Geometrically, $f'(a)$ is the slope of the tangent line. In an application, it gives the local rate at that instant rather than an average over a finite interval.

The sign tells the direction of change

A positive rate means the output is increasing as the input increases. A negative rate means it is decreasing. A zero derivative means the instantaneous rate is zero, though that alone does not guarantee a maximum or minimum.

For example, if a temperature function rises from 18°C to 24°C over three hours, its average rate of change over that interval is

$$\frac{24-18}{3}=2\text{ °C/hour}.$$

The instantaneous rate at a particular time could be higher, lower, or zero depending on the shape of the temperature curve.