IB Maths AA SLTopic 5 — CalculusPaper 1 & 2~10 min read
Introduction to Derivatives
When a car speeds up, you’re watching calculus in action. The car’s position changes over time — and the rate at which it changes is its speed. Calculus is just the maths that describes how things change. The big tool you’ll learn is the derivative — a function that tells you exactly how fast something is changing at any moment.
📘 What you need to know
Calculus is the maths of change and rates.
The gradient of a curve at a point = the gradient of the tangent at that point.
A derivative is a function that gives you the gradient at any value of x.
The derivative is built using limits: imagine sliding a chord toward a tangent — its gradient gets closer and closer to the tangent’s gradient.
Notation: if y = f(x), the derivative is written dydx or f′(x). Both mean exactly the same thing.
You can use different variables too: e.g. if V = f(s), the derivative is dVds.
What’s calculus actually about?
Calculus has a reputation for being scary, but at its core, it answers one simple question: “How quickly is something changing?” Whenever a quantity isn’t constant — speed, temperature, the height of water in a tank, a stock price — calculus gives you the tools to describe and predict that change.
You see rates of change everywhere:
🚗
Speed
how fast position changes over time
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Acceleration
how fast speed itself changes
🌡️
Cooling
how fast a hot drink loses heat
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Filling rate
how fast a tank is filling up
When the speedometer in a car says “60 km/h”, that’s a derivative in action. It’s not measuring distance — it’s measuring how quickly distance is changing right at this moment. That’s exactly what a derivative does for any function.
From straight lines to curves
You already know how to find the gradient of a straight line: pick two points and use rise over run. The thing is, a straight line has the same gradient everywhere — it doesn’t change.
Curves are different. Look at the graph of y = x2: at x = −1 it’s sloping down, at x = 0 it’s flat, and at x = 2 it’s sloping up steeply. The gradient changes from point to point. So we need a way to find the gradient at one specific point.
Linear vs curved — gradient changes for curves
The tangent: gradient at a single point
To find the gradient of a curve at one specific point, we draw a tangent there — a straight line that just touches the curve at that point without cutting through. The gradient of that tangent line is what we call the gradient of the curve at that point.
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Big idea: the gradient of a curve at a point = the gradient of its tangent there
This single sentence is the foundation of all calculus. Once we can describe how to find that tangent’s gradient, we can describe how anything is changing.
How do we actually find a tangent’s gradient?
Here’s the clever trick that makes calculus work. We can’t directly measure a tangent’s gradient with two points, because the tangent only touches the curve at one point. But we can easily find the gradient of a chord — a straight line joining two points on the curve.
So we cheat. We pick a second point Q close to our point of interest P, draw the chord PQ, and find its gradient. Then we slide Q closer and closer to P. As Q gets closer, the chord starts to look more and more like the tangent. The gradient of the chord gets closer and closer to the gradient of the tangent.
Sliding Q toward P — chords approach the tangent
🤔 Why does this work?
Imagine standing on a curve and looking along the chord PQ. When Q is far away, you see a steep slope cutting through. As Q creeps closer, the chord rotates — eventually, it lies almost flat against the curve, pointing in the same direction as the tangent. The gradient of the chord and tangent become indistinguishable. That “becoming indistinguishable” is what mathematicians call a limit.
What’s a limit?
A limit is the value a function gets closer and closer to, even if it never actually reaches it. It’s a way to talk about behaviour at a point where you can’t just plug in directly.
For example, the function f(x) = x4 − 1x − 1 is undefined at x = 1 (you’d be dividing by zero). But if you plug in values close to 1, the output approaches 4. So we say the limit of f(x) as x → 1 is 4.
x
0.9
0.99
0.999
1.001
1.01
1.1
f(x)
3.439
3.940
3.994
4.006
4.060
4.641
Notice how f(x) approaches 4 from both sides as x approaches 1. That’s the limit.
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The link between limits and derivatives
The gradient of the tangent at P is the limit of the chord gradients as Q slides toward P. That limit IS the derivative. Every derivative you’ll ever calculate is built on this idea.
So what is a derivative?
A derivative is a function that gives you the gradient of the original function at any value of x. You input an x-value, and it outputs the gradient at that point.
In plain English
derivative = gradient function
So if the original function f(x) describes a car’s position over time, the derivative f′(x) describes the car’s speed at any time. Same input, different output. The derivative is its own function — built from the original.
In the next note we’ll learn the rules for finding derivatives quickly without doing the chord-sliding thing every time. But the idea — gradient at a point as the limit of chord gradients — is the foundation everything else stands on.
The notation: two ways of writing the same thing
You’ll see two notations for the derivative. Both mean exactly the same thing — different mathematicians just preferred different styles.
Leibniz notation
dydx
“the derivative of y with respect to x”
Lagrange (prime) notation
f′(x)
“f-prime of x”
So if y = f(x), then dydx = f′(x). They’re identical — just two ways of writing the gradient function.
Different variables work the same way
The letters don’t have to be x and y. If a function uses different variables, the notation changes to match:
If V = f(s) (volume in terms of length), then dVds = f′(s).
If P = g(t) (population over time), then dPdt = g′(t).
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“d top, d bottom — top changes with respect to bottom”
dydx means “how does y change when x changes”. dVds means “how does V change when s changes”. The fraction-style notation is actually telling you exactly that — the change in the top variable per unit change in the bottom one.
Worked examples
WE 1
Find the gradient of a chord
The graph of y = f(x) where f(x) = x³ − 2 passes through the points P(2, 6), A(2.3, 10.167), B(2.1, 7.261), and C(2.05, 6.615125). Find the gradient of the chords [PA], [PB], and [PC].
Use the gradient formula: m = (y₂ − y₁)/(x₂ − x₁) for each chord.chord PAm = (10.167 − 6)/(2.3 − 2) = 4.167/0.3= 13.89chord PBm = (7.261 − 6)/(2.1 − 2) = 1.261/0.1= 12.61chord PCm = (6.615125 − 6)/(2.05 − 2) = 0.615125/0.05= 12.3025PA: 13.89 · PB: 12.61 · PC: 12.30notice how the gradients are getting smaller and closer to a single value as the points move closer to P!
WE 2
Estimate the gradient of the tangent
Using the chord gradients from WE 1 (13.89, 12.61, 12.30), estimate the gradient of the tangent to the curve y = x³ − 2 at the point P(2, 6).
As the second point gets closer to P, the chord gradient approaches the tangent gradient. Look at the trend.Chord gradients:13.89 → 12.61 → 12.30…As Δx shrinks, the gradient approaches a limit just above 12.tangent gradient ≈ 12the actual answer is exactly 12 (since f'(x) = 3x², so f'(2) = 12) — but at this stage we just estimate from the trend!
WE 3
Estimate a limit from a table
The function g(x) is given by the values below. Estimate the limit of g(x) as x approaches 2.
x
1.9
1.99
1.999
2.001
2.01
2.1
g(x)
4.61
4.96
4.996
5.004
5.04
5.41
Look at what value g(x) approaches as x gets closer to 2 from both sides.From the left:4.61 → 4.96 → 4.996 → approaching 5From the right:5.004 → 5.04 → 5.41… approaching 5 toolimit ≈ 5both sides agree → the limit exists and equals 5!
WE 4
Translate between notations
For each function below, write the derivative using both notations.
(a) y = f(x) (b) A = g(r) (area in terms of radius) (c) h = p(t) (height in terms of time)
Match Leibniz “d top / d bottom” with Lagrange “function-prime of input”.part (a)y depends on x:dy/dx or f'(x)part (b)A depends on r:dA/dr or g'(r)part (c)h depends on t:dh/dt or p'(t)notation matches the variables in the functiond(output) / d(input) — that’s the pattern!
WE 5
Interpret a derivative in context
The volume of water (in litres) in a tank at time t (seconds) is given by V = f(t). At t = 10, the gradient of the tangent to the graph of V against t is 0.8. Interpret this in context.
The gradient of V against t gives the rate of change of volume with time.f'(10) = 0.8 means dV/dt = 0.8 at t = 10.Units: litres per second.at t = 10s, the tank is filling at 0.8 L/salways include units when interpreting a derivative in context!
💡 Top tips
“Derivative” and “gradient function” mean the same thing. Use whichever phrasing the question uses.
The derivative is a function, not a number. Plug in an x-value to get the gradient at that specific point.
Sketch the curve and tangent when you’re getting a feel for derivatives. Visual intuition makes the algebra easier later.
Use chord gradients to estimate when asked: pick close points either side of x, find the chord gradient, and look at the trend.
For limits, check both sides — values approaching from below AND from above. If they agree, that’s the limit.
Leibniz and Lagrange notation are interchangeable. Don’t get thrown by which one a question uses.
In real-world contexts, dy/dx has units of (y-units) per (x-unit). Always state them.
Smaller Δx → better estimate for the tangent gradient. Closer points give more accurate chord gradients.
⚠ Common mistakes
Confusing chord with tangent. A chord cuts through two points; a tangent touches at one. Different things, related by limits.
Treating dy/dx as a fraction in this context. It’s notation, not division — though it does behave like a fraction in some advanced techniques.
Saying the derivative is a number. It’s a function. f'(x) gives a number only after you plug in a specific x.
Forgetting that the gradient changes on a curve. Unlike a line, a curve’s gradient is different at every point.
Using the wrong variable letters. If V = f(s), the derivative is dV/ds — not dy/dx.
Mixing up “rate of change” with “amount”. A derivative tells you how fast something is changing, not how much there is.
Forgetting units in real-world problems. A derivative always has units of (output) per (input).
Trying to plug a value where the function is undefined. You need a limit there, not a direct substitution.
Now you understand what a derivative is — a gradient function built from the limit of shrinking chord gradients. The next note shows you the actual rules for differentiating powers of x — so you can find derivatives in seconds without ever sliding chords around again.
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