Key Idea: This topic is about reading the steepness of a curve: the derivative is the gradient at a single point and the instantaneous rate of change. It's the conceptual foundation for all of calculus — mostly tested on Paper 1 (interpret and reason, no calculator).
📈 What the derivative is
A straight line has one gradient everywhere; a curve keeps changing steepness, so we use the gradient at a point — the gradient of the tangent there (never a chord). The same number is also a rate of change: how fast y changes per unit of x at an instant (if x is time, the instantaneous rate). Always carry units — (y-units) per (x-unit).
✍️ Notation: f′(x) and dy/dx
- "f prime of x" — the gradient function
- rate of change of y with respect to x (same thing)
± Sign of the gradient = shape of the curve
| Sign of f′(x) | What the curve is doing |
|---|---|
| f′(x) > 0 | Increasing — going up as x increases |
| f′(x) < 0 | Decreasing — going down as x increases |
| f′(x) = 0 | Stationary — momentarily flat (peak, trough or flat spot) |
✏️ IB-style worked examples
IB-style question — interpret a rate of change
The height h (cm) of a plant after t weeks satisfies dh/dt = 3.5 when t = 6. Interpret this value, with units.
Step by step:
dh/dt is the rate the height changes per week.
The value is given at t = 6 specifically.
At t = 6 weeks the plant's height is increasing at 3.5 cm per week.
IB-style question — gradient at a point from f′(x)
The gradient function of a curve is f′(x) = 4x − 5. Find the gradient of the curve at x = 2.
Step by step:
Substitute x = 2 into the gradient function.
Evaluate.
The gradient at x = 2 is 3.
IB-style question — increasing or decreasing?
For a function g, g′(−1) = 7 and g′(4) = −2. State whether g is increasing or decreasing at each point.
Step by step:
At x = −1 the derivative is positive.
At x = 4 the derivative is negative.
Increasing at x = −1; decreasing at x = 4.
Important: The gradient at a point is the gradient of the tangent there — not the average gradient of a chord between two points, and not a single fixed number for the whole curve. To get a value, substitute the x into f′(x).
Tap each card to reveal the answer.
What does dy/dx represent? The gradient of the curve at a point / the rate of change of y with respect to x.
f′(x) < 0 at a point — what's the curve doing? Decreasing (going downhill as x increases).
f′(x) = 0 at a point — what kind of point is it? A stationary point — momentarily flat (peak, trough or flat spot).
dV/dt = −8 L/min for a draining tank — meaning? The volume is decreasing at 8 litres per minute.
Exam tips
- Gradient at a point = gradient of the tangent there (not a chord, not one number for the whole curve).
- f′(x) and dy/dx mean the same thing — the gradient function.
- To get a gradient value, substitute the x-value into f′(x).
- f′(x) > 0 increasing, f′(x) < 0 decreasing, f′(x) = 0 stationary.
- A rate of change carries units: (y-units) per (x-unit), e.g. cm per week.