What is a derivative?
Average change uses two points. A derivative asks what the slope approaches as those two points get closer together.
From average slope to slope right now
Take f(x) = x² at x = 3. Choose a nearby point at x = 3 + h. The average slope between the two points is:
[f(3 + h) − f(3)] / h = [(3 + h)² − 9] / h = 6 + h, for h ≠ 0.
As h moves toward 0, those secant slopes approach 6 from both sides. Therefore f′(3) = 6. We do not divide by zero; we take the limit of slopes for nearby nonzero h.
| h | −0.1 | −0.01 | 0.01 | 0.1 |
|---|---|---|---|---|
| 6 + h | 5.9 | 5.99 | 6.01 | 6.1 |
What does the number mean?
If x measures seconds and f(x) measures meters, f′(3) = 6 means the instantaneous velocity at 3 seconds is 6 meters per second. A derivative can be positive, negative, or zero; its sign tells the direction of change. The same limit idea behind a limit is what turns nearby average slopes into an instantaneous slope.
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