What does a derivative actually tell you?
A derivative tells you how fast a quantity is changing at a single instant: the slope of the tangent line to its graph at that point. You find it by taking the slope between two nearby points and letting the gap between them shrink to zero.
Watch it explained
What does a derivative actually tell you?
Starts from a car's speedometer and a graph of distance over time, shows why an average speed misses the detail, then shrinks the gap between two points to zero until the secant line becomes the tangent line, whose slope is the derivative.
Scenes, in order:
- Your speedometer is a derivative
- Distance over time
- Average speed misses details
- Shrink the gap to zero
- Slope of tangent = derivative
From average rate to instantaneous rate
Between two points on a graph, the average rate of change is the slope of the secant line through them: Δy/Δx. If the points are x and x + h, that slope is [f(x + h) − f(x)] / h. An average hides what happens in between, the way an average speed for a whole trip hides every stop and every stretch of highway.
Slide the second point toward the first and h shrinks. The secant line swings toward the tangent line, and its slope settles on a single number. That limit is the derivative: f′(x) = lim (h → 0) of [f(x + h) − f(x)] / h. A speedometer reading is exactly this: the derivative of distance with respect to time.
A worked example: f(x) = x² at x = 3
Take secant slopes from x = 3 with smaller and smaller gaps:
- 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 slopes close in on 6. The algebra confirms it: [(3 + h)² − 9] / h = (6h + h²) / h = 6 + h, which goes to 6 as h goes to 0. Doing the same with a general x gives f′(x) = 2x. Units carry through: if f is in meters and x in seconds, f′ is in meters per second.
Reading derivatives on exams
- f′(x) > 0 means f is increasing there; f′(x) < 0 means it is decreasing.
- f′(x) = 0 means a horizontal tangent, which is not always a maximum or minimum: x³ has f′(0) = 0 and keeps rising.
- A function can be continuous and still have no derivative at a point, like |x| at its corner at 0.
- When matching graphs, read the height of the f′ graph as the slope of the f graph, not the height of f.
- Always state units in context: "the tank is filling at 4 liters per minute at t = 2."
The film on this page
The Isotoon video here answers "What does a derivative actually tell you?" It starts from a speedometer, plots distance over time, shows why an average speed misses detail, then shrinks the gap between two points until the secant becomes the tangent. Your first video is free, and this question is one of the examples you can pick. Next, see how derivatives combine in the chain rule and how they undo integrals in the fundamental theorem of calculus. Course context is on AP Calculus and calculus.
Questions
- Is a derivative the same as a slope?
- At a single point, yes: the derivative is the slope of the tangent line there. As a function, f′(x) gives that slope at every x, so it changes as you move along the curve.
- Why can't you just plug h = 0 into the difference quotient?
- Because you would be dividing 0 by 0. The limit asks what value the slope approaches as h gets close to 0, which is why you simplify first, as in (6h + h²) / h = 6 + h, and then let h go to 0.
- What are the units of a derivative?
- The units of the output divided by the units of the input. If volume is in liters and time is in minutes, the derivative of volume with respect to time is in liters per minute.