The big idea: A straight line has the same gradient everywhere — you can measure it once and you are done. A curved function like y = x² has a different gradient at every single point. Differentiation is the tool that finds the gradient of a curve at any point you choose.
Straight lines — gradient is constant
Take the line y = 3x + 1. No matter where you are on it, the slope is always 3. Pick x = 0, x = 5, x = −10 — the gradient is always 3.
Curves — gradient changes at every point
Now take y = x². The curve is flat near x = 0, rises gently for small x, and rises steeply for large x.
| Position on y = x² | Is the curve steep or flat? | Gradient (we will calculate this soon) |
|---|---|---|
| x = 0 | Flat — bottom of the curve | 0 |
| x = 1 | Gently rising | 2 |
| x = 3 | Rising more steeply | 6 |
| x = −2 | Falling steeply | −4 |
You can already see that there is no single answer for the gradient of y = x². It depends on where you are. That is the problem differentiation solves.
The tangent line idea: Imagine placing a ruler so it just touches the curve at one point without crossing it. That ruler is the tangent line. The gradient of the tangent line is the gradient of the curve at that point. Differentiation finds that gradient without having to draw anything.
| Function type | Gradient behaviour | Do you need differentiation? |
|---|---|---|
| Linear (y = mx + c) | Constant — always m | No — read the gradient straight from the equation |
| Curved (y = x², y = x³, …) | Different at every point | Yes — differentiation gives the gradient at any x |
[Diagram: math-derivative-tangent] - Available in full study mode
The big idea: The derivative of a function is a new function. When you plug an x-value into the derivative, it tells you the gradient of the original curve at that x. The process of finding the derivative is called differentiation.
How to write the derivative
IB uses two main notations — you need to recognise both.
| Notation | How you read it | Example |
|---|---|---|
| f′(x) | f prime of x | If f(x) = x², then f′(x) = 2x |
| dy/dx | dee y by dee x | If y = x², then dy/dx = 2x |
| y′ | y prime (less common in AI SL) | y′ = 2x |
f′(x) and dy/dx mean the same thing: Both tell you the gradient of the function at any x. Use whichever notation the question uses. IB problems tend to use f′(x) when they write the function as f(x) = ..., and dy/dx when they write it as y = ...
What differentiation gives you
Start with a function f(x). Differentiate it. You get f′(x) — a new function.
The pattern — preview (we will calculate this properly in 5.1.2)
f(x) = x². What is f′(x)?
Step by step
- The derivative of x² turns out to be 2x.
- Now plug in x = 3 to find the gradient at x = 3.
- This confirms the table from Section 1. At x = 3 the curve y = x² has gradient 6.
Final answer
f′(3) = 6. The gradient of y = x² at x = 3 is 6.
Read the question carefully: If IB asks for f′(x), they want the gradient function — leave it in terms of x. If they ask for the gradient at x = 2, substitute x = 2 into f′(x) to get a number.
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The big idea: Gradient measures how fast something changes. When the function models a real situation, the derivative is the rate of change — how quickly the output is changing per unit of input. This is one of the most important ideas in IB calculus.
From gradient to rate of change
If y represents height (in metres) and x represents time (in seconds), then dy/dx represents metres per second — the rate at which height changes with time. That is velocity.
| f(x) represents | f′(x) represents | Units of f′(x) |
|---|---|---|
| Height h (metres) at time t (seconds) | Rate of change of height — vertical velocity | metres per second |
| Population P at time t (years) | Rate of change of population | people per year |
| Revenue R (dollars) when x units sold | Extra revenue per extra unit sold — marginal revenue | dollars per unit |
| Temperature T (°C) at time t (hours) | Rate of temperature change | °C per hour |
Worked context example: A ball is thrown upward. Its height (in metres) at time t seconds is given by h(t) = 20t − 5t². Find h′(t) and interpret h′(2). (We will differentiate properly in 5.1.2 — for now, accept that h′(t) = 20 − 10t.)__LINEBREAK__h′(t) = 20 − 10t__LINEBREAK__h′(2) = 20 − 10(2) = 0__LINEBREAK__At t = 2 seconds the ball is momentarily not moving upward or downward — it is at the top of its path.
Positive, negative, or zero rate of change: f′(x) > 0 means the quantity is increasing at that point.__LINEBREAK__f′(x) < 0 means the quantity is decreasing at that point.__LINEBREAK__f′(x) = 0 means the quantity is momentarily not changing — a stationary moment.
IB context questions always want units: When IB asks you to interpret a derivative in context, your answer must include the units. 'The rate of change is 4' earns no credit. 'The temperature is increasing at 4 °C per hour' earns full credit.
The big idea: You can tell a lot about f′(x) just by looking at the graph of f(x). Where the curve rises, f′(x) is positive. Where it falls, f′(x) is negative. Where it flattens out (top of a hill or bottom of a valley), f′(x) = 0.
| What the graph looks like at x | Sign of f′(x) | What that means |
|---|---|---|
| Curve is rising steeply to the right | f′(x) is large and positive | Function increasing fast |
| Curve is rising gently to the right | f′(x) is small and positive | Function increasing slowly |
| Curve is flat (top of a hill) | f′(x) = 0 | Stationary point — local maximum |
| Curve is flat (bottom of a valley) | f′(x) = 0 | Stationary point — local minimum |
| Curve is falling gently to the right | f′(x) is small and negative | Function decreasing slowly |
| Curve is falling steeply to the right | f′(x) is large and negative | Function decreasing fast |
Worked graph-reading example: A sketch of f(x) shows that the curve rises from left to right until it reaches a peak at x = 2, then falls.__LINEBREAK___• For x < 2: the curve is rising → f′(x) > 0 • At x = 2: the curve is at its peak (flat at the top) → f′(2) = 0 • For x > 2: the curve is falling → f′(x) < 0
Common confusion — f(x) vs f′(x): Students often mix up the value of the function with the gradient. If the curve is high up on the y-axis but flat, f(x) is large but f′(x) = 0. The height of the curve tells you f(x). The steepness and direction of the curve tells you f′(x).
Quick graph-reading practice
A function f has a graph that rises from x = 0 to x = 3, then falls from x = 3 to x = 7. State the sign of f′(1), f′(3), and f′(6).
Step by step
- At x = 1, the curve is rising.
- At x = 3, the curve is at its peak — flat.
- At x = 6, the curve is falling.
Final answer
f′(1) > 0, f′(3) = 0, f′(6) < 0.