Unit 12 · Calculus
Differentiation
Nine lessons that take you from never having seen a derivative to answering any differentiation question on either paper. Every lesson teaches, then makes you practise until it is automatic.
The nine lessons
Work through them in order. Each one is about ninety minutes and stands on its own page, so you can open one, finish it, and print it as a worksheet.
Gradient of a chord, shrinking it to a tangent, the notation, then the six standard derivatives.
02Chain, product and quotient, one at a time, each with its own full practice set.
03Finding the equation of a tangent and of a normal at a given point.
04Finding them, and deciding their nature by both the first and the second derivative test.
05Turning a worded problem into one variable, then optimising it.
06When one quantity changes because another does, and the chain rule links them.
07Approximating a change in one quantity from a small change in another.
08Displacement, velocity and acceleration for motion in a straight line.
09Recognising which application a question wants, the six checks, and a large mixed set.
What Unit 12 asks of you
You need to be able to
- Understand what a derived function is, and limits informally
- Use the notation \(f'(x)\), \(f''(x)\), \(\dfrac{dy}{dx}\), \(\dfrac{d^{2}y}{dx^{2}}\), \(\delta x\)
- Differentiate \(x^{n}\) for any rational \(n\), and \(\sin x\), \(\cos x\), \(\tan x\), \(e^{x}\), \(\ln x\)
- Handle constant multiples, sums, and composite functions
- Use the product and quotient rules
- Find gradients, tangents and normals
- Find stationary points and decide their nature
- Do connected rates of change, and small increments
- Solve practical maximum and minimum problems
- Work with displacement, velocity and acceleration
You are not asked for
- Differentiation from first principles
- Any formal treatment of limits
- Points of inflexion
- Implicit or parametric differentiation
- Trigonometric derivatives in degrees
That list is worth reading twice. Every year candidates spend time in the exam on work the syllabus never asks for. None of it earns a mark.
The two papers
| Paper | Time | Marks | Calculator | Weight |
|---|---|---|---|---|
| Paper 1 | 2 hours | 80 | Not allowed | 50% |
| Paper 2 | 2 hours | 80 | Scientific calculator | 50% |
There are no tiers. Grades run from A* to E. 0606 is normally taken alongside 0580, and it assumes you already have the algebra from 0580 Extended.
Angles are always in radians when you differentiate a trigonometric function. A calculator left in degrees has cost more marks in this topic than any other single mistake.
How to use this
Each lesson is built the same way, and the order matters. Do not skip to the practice — the practice only works because of what comes before it.
| Part | What it is | What to do with it |
|---|---|---|
| Worked example | A complete solution with every line shown. | Read it with a pen. Copy it out. |
| Guided steps | The same kind of question with the working half-written. | Fill the blanks. Use the hint only after trying. |
| Fluency drill | Ten to twenty short questions, marked instantly. | Until it is fast and boring. |
| Exam-style | Full questions with the marks shown. | Write the whole solution out, then reveal mine. |
| Error hunt | A finished solution with one mistake in it. | Find the line. This is how you learn to check your own work. |
Threaded through every lesson are marks notes, in red. Each one is a place where candidates who understood the mathematics still lost marks. They are short. Read every one.