The Derivative as a Rate of Change
Simple Explanation
The AVERAGE rate of change of f between x=a and x=b is [f(b)−f(a)]/(b−a) — the slope of the secant line through those two points. Shrinking the interval [a,b] down toward a single point gives the INSTANTANEOUS rate of change at that point — the derivative — which is the slope of the tangent line there.
Why Do We Need It?
This is the bridge between something concrete and measurable (a secant slope over an interval) and the more abstract idea of an instantaneous rate (a tangent slope at a single point).
See It
A parabola y=x squared, with a straight secant line drawn through the two points (0,0) and (2,4) on the curve
Formula
Average Rate of Change
Average rate of change = [f(b) − f(a)] / (b − a)
The average rate of change of a function f between x=a and x=b is the slope of the secant line connecting the two points (a, f(a)) and (b, f(b)).
- a, b
- — the two x-values marking the interval
- f(a), f(b)
- — the function's values at those two x-values
When to use it: Whenever you need the average rate of change of a function over an interval, before narrowing that interval down to an instantaneous rate.
Worked Example
Find an average rate of change
A ball's height is given by h(t) = t² (in meters, for t in seconds). Find the average rate of change of height between t=0 and t=2.
Why Does This Work?
The average rate of change is exactly the slope formula (rise over run) applied to the two points (a, f(a)) and (b, f(b)) on the curve — as those two points are brought closer and closer together (b→a), the secant line through them rotates to become the tangent line, and its slope becomes the instantaneous rate of change.
Real-Life Example
Average speed versus marginal production cost
A factory tracks its average cost per unit over a range of production levels, but managers ultimately want to know the cost of producing exactly ONE more unit at the current level.
The average cost over a range is a secant slope; the "marginal cost" at an exact production level is the derivative — the instantaneous rate of change.
Practice
Given f(x) = x², find the average rate of change of f between x=1 and x=4.
MediumCommon mistake
Confusing the average rate of change (a secant slope, over an interval) with the instantaneous rate of change (a tangent slope, at a single point) — they are related but generally different numbers.
Quick Review
- Average rate of change = [f(b)−f(a)]/(b−a), the secant line's slope.
- Shrinking the interval to a single point gives the instantaneous rate — the derivative.
- Instantaneous rate of change = the slope of the tangent line at that point.