TL;DR:
The derivative is the rate of change at a single instant: the slope of the tangent line.
It is the limit of average rates over shorter and shorter intervals.
Its units are output units per input unit, such as metres per second.
A car's speedometer shows a number at one instant. But speed is distance divided by time, and an instant has no time in it. How can a single moment have a speed? Calculus answers by squeezing the time interval smaller and smaller and watching what the average speed approaches.
Average rate of change
Between two moments, the average rate of change is the change in position divided by the change in time. It is the slope of the straight line joining two points on a graph. If a car goes 120 km in 2 hours, its average speed is 120 / 2 = 60 km/h. That tells you nothing about the speed at any instant, because it may have stopped and sprinted.
Worked example: speed from a table of positions
A ball rolls down a ramp, and its distance from the top is s = 5t² metres after t seconds (an example model). The table gives positions.
t (s) | s (m) |
|---|---|
0 | 0 |
1 | 5 |
2 | 20 |
3 | 45 |
Average speed from 1 s to 3 s: (45 − 5) / (3 − 1) = 40 / 2 = 20 m/s. From 1 s to 2 s it is (20 − 5)/1 = 15 m/s, and from 2 s to 3 s it is (45 − 20)/1 = 25 m/s. The speed changes within the interval, so the averages differ.
To get the speed at exactly t = 2, shrink the interval that starts at 2.
Interval | Change in s | Average speed (m/s) |
|---|---|---|
2 to 3 | 25 | 25 |
2 to 2.1 |
| 20.5 |
2 to 2.01 |
| 20.05 |
2 to 2.001 |
| 20.005 |
The averages 25, 20.5, 20.05, 20.005 are heading for 20. The speed at the instant t = 2 is 20 m/s. That value is the derivative of s at 2. Using calculus rules, the derivative of 5t² is 10t, and at t = 2 it equals 20. The next articles explain that rule.
The tangent line
On the graph of s against t, each average speed is the slope of a line through two points on the curve. As the second point slides toward the first, that line pivots into the tangent line, which just touches the curve. The derivative is the slope of the tangent. Here the tangent at t = 2 has slope 20 and passes through (2, 20), so its equation is s = 20 + 20(t − 2). Check it at t = 2.1: the tangent gives 22, the curve gives 22.05. Very close, and closer the nearer you get to 2.
Units
The derivative has units of the output divided by the input. Position in metres, time in seconds, so the derivative is in metres per second. If a tank's volume in litres is a function of minutes, its derivative is litres per minute. A units check is the quickest way to catch a mistake: an answer in metres is not a speed.
The sign matters too. A positive derivative means the quantity is increasing at that instant. A negative one means it is decreasing. Zero means it is momentarily flat, as at the top of a thrown ball's path.
Try one yourself: s = t² at t = 3
Take a second model, s = t² metres. At t = 3, s = 9. Shrink the interval: from 3 to 3.1, s = 9.61, so the average speed is (9.61 − 9) / 0.1 = 6.1 m/s. From 3 to 3.01, s = 9.0601, so the average speed is 0.0601 / 0.01 = 6.01 m/s. The values head for 6 m/s. The derivative rule for t² is 2t, and 2 × 3 = 6, which agrees. Notice that the pattern 6.1, 6.01 shows the error shrinking by a factor of ten each time we shrink the interval by ten.
How to check your answer
Order. The averages over shrinking intervals should settle toward one number, from one or both sides.
Bracket. The speed at 2 s should lie between the average over 1 s to 2 s (15) and the average over 2 s to 3 s (25). It does: 20.
Units. m divided by s gives m/s.
Shape. The ball speeds up, so the slope of the curve should grow along the table. It does.
Common mistakes
Confusing the average rate over a long interval with the instantaneous rate.
Putting the time change on the wrong side. Divide by the change in time.
Rounding the tiny intervals so much that every average looks the same.
Forgetting the units, or giving the derivative in the units of position.
Thinking the derivative is the value of the function. Position 20 m and speed 20 m/s happen to match here by coincidence at t = 2.
Where you will use it next
The next step is a rule that finds derivatives without tables, in the power rule and where it comes from. Derivatives of repeating patterns link back to arithmetic and geometric sequences, where the slope between neighbouring terms is the common difference. Anything that changes smoothly, from temperature to population to the height of a ball, has a derivative.
Frequently asked questions
Is the derivative the same as the slope?
The derivative at a point is the slope of the tangent line there. For a straight line, it is the same everywhere. For a curve, it changes from point to point.
Can I compute a derivative from a table alone?
Only an estimate. The closer the table's inputs are, the better the estimate, which is why shrinking intervals matter.
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