FreeLesson 1 of 36 min read
What a derivative measures
Learn how the derivative turns the slope between two points into the exact rate of change at a single instant.
Almost everything interesting changes. A car speeds up, a population grows, a tank drains. Knowing a value is useful, but knowing how fast it is changing at one moment is often more useful. The derivative is the tool that measures this instant rate of change, and it underpins the rest of calculus.
Start with the average rate of change
Suppose a quantity y depends on x through a function f. Between x = a and x = b, the average rate of change is the change in output divided by the change in input: (f(b) - f(a)) / (b - a). On a graph, this is the slope of the straight line joining the two points. That line is called a secant line.
The trouble is that an average hides detail. A car that covers 100 km in 2 hours averages 50 km/h, but at any given moment it might be going 30 km/h or 80 km/h. To find the speed at one instant, we need to shrink the time interval.
Shrink the gap
Fix a point x and move a small step h away from it. The average rate of change over that small step is (f(x + h) - f(x)) / h. As h gets closer and closer to 0, the secant line pivots towards the line that just touches the curve at x: the tangent line. The value this ratio approaches is the derivative, written f'(x).
So the derivative at a point is all of these at once:
- The limit of (f(x + h) - f(x)) / h as h approaches 0.
- The slope of the tangent line to the graph at that point.
- The instantaneous rate of change of f with respect to x.
- A number with units: output units per input unit.
Example: the derivative of f(x) = x^2 at x = 3
First, try small steps numerically:
h = 1: (16 - 9) / 1 = 7
h = 0.1: (9.61 - 9) / 0.1 = 6.1
h = 0.01: (9.0601 - 9) / 0.01 = 6.01
The values seem to approach 6.
Now confirm with algebra:
f(3 + h) = 9 + 6h + h^2
f(3 + h) - f(3) = 6h + h^2
Divide by h: 6 + h
As h approaches 0, this approaches 6.
So f'(3) = 6. The tangent line at (3, 9) has slope 6.
If you repeat the algebra at a general point x, you get (2xh + h^2) / h = 2x + h, which approaches 2x. So the derivative of x^2 is 2x. The same limit idea gives the power rule: the derivative of x^n is n x^(n-1). You will use this rule constantly in later lessons.
Always read the units
Example: velocity from position
A ball's position is s(t) = 5t^2 metres after t seconds.
Using the power rule, s'(t) = 10t.
At t = 2: s'(2) = 20.
Check numerically: (s(2.1) - s(2)) / 0.1 = (22.05 - 20) / 0.1 = 20.5, close to 20.
Meaning: at exactly 2 seconds the ball is moving at 20 metres per second.
Notice that the units are metres per second because we divided metres by seconds. A derivative always carries the units of the output divided by the units of the input. A negative derivative means the quantity is decreasing at that instant; a zero derivative means it is momentarily neither rising nor falling.
Next time you meet a derivative, picture the secant line shrinking into a tangent, and say its meaning aloud with units, such as 'litres per minute at t = 5'. That habit will make every later topic easier.
Key takeaways
- The derivative is the limit of average rates of change as the interval shrinks to zero.
- Geometrically, it is the slope of the tangent line at a point.
- Its units are output units per input unit, and its sign shows whether the quantity is rising or falling.
Check your understanding
Written with AI help. Spotted a mistake? Tell us.