Free IB Maths AA HL lesson: Derivative from first principles
A ready-to-teach IB Maths AA HL lesson on Derivative from first principles (Unit 5 · Calculus): editable slides and teacher notes, free; a printable worksheet in 2 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans3 questions, 15 marks: Fluency, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Differentiation from First Principles
- Finding Constants for a Smooth Join
Key ideas on the slides
- The derivative is the limit of the gradient of a chord: f′(x) = lim_(h → 0)((f(x + h) − f(x))/h).
- Example: f(x) = x²: ((x + h)² − x²)/h = (2xh + h²)/h = 2x + h → 2x as h → 0.
- Average rate of change from x = a to x = b is (f(b) − f(a))/(b − a) (a chord); the instantaneous rate of change is f′(a) (a tangent).
- In a first-principles proof, write the limit notation on every line until h is removed, and do not let h = 0 before dividing.
By the end of the lesson, students should be able to answer a question like: Use first principles to find the derivative of f(x) = x^2.
Key vocabulary
- limit: The value an expression approaches as the variable approaches a value.
- chord: A straight line joining two points on a curve.
- derivative: f′(x): the gradient function, the instantaneous rate of change.
Common mistakes this lesson tackles
- Substituting h = 0 before simplifying, getting 0/0.
- Dropping "lim_(h → 0)" from the working.
- Confusing average and instantaneous rates of change.
The worked example
Use the formal definition of the derivative from first principles to prove that if f(x) = x³, then f′(x) = 3x².
- State the formal limit definition of the derivative: f′(x) = lim_(h → 0) (f(x+h) − f(x))/h.
- Substitute the specific function into the formula: f′(x) = lim_(h → 0) ((x+h)³ − x³)/h.
- Expand the binomial cubic correctly: f′(x) = lim_(h → 0) (x³ + 3x²h + 3xh² + h³ − x³)/h.
- Cancel the x³ terms in the numerator: f′(x) = lim_(h → 0) (3x²h + 3xh² + h³)/h.
- Factor out and cancel the common h term: f′(x) = lim_(h → 0) (3x² + 3xh + h²).
- Evaluate the limit by directly substituting h=0: 3x² + 3x(0) + (0)² = 3x².
Tip: Never drop the "lim_(h → 0)" notation in your working steps until the very final line when you actually evaluate the limit by setting h=0. Dropping it early is a notation error penalized by examiners.
Calculator tips
- The d/dx key checks the value at a point; a table of (f(a + h) − f(a))/h for small h shows the limit.
Where this lesson fits
Derivative from first principles is in Unit 5 · Calculus of the IB Maths AA HL scheme of work.
- The lesson before: Bayes' theorem (scheme-of-work lessons: Applications & Problem Solving; Problems with Three Events)
- The next lesson: Implicit differentiation (scheme-of-work lessons: Implicit Differentiation; Tangents & Normals to Implicit Curves)
Lesson plan
| 0–5 min | Do now: students write down what they remember about Informal limits and Bayes' theorem |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (3 questions in 2 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA HL Unit 5: First Principles practice questions.
With a school plan
Set Derivative from first principles as homework marked by AI, print this lesson's levelled worksheet with answers, and rebuild the lesson for your class with Create lesson. Free for 30 days as a teacher, no card.
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