Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Differentiate from first principles
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Differentiate from first principles.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Estimate the gradient of a curve with chords; Evaluate limits by substitution or by factorising
Leads to: Differentiate axⁿ with the power rule
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Estimate the gradient of a curve with chords; Evaluate limits by substitution or by factorising). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- For y = x², find the gradient of the chord from x = 0 to x = 0.1. → 0.1Prompt: Gradient = change in y ÷ change in x.
- Find lim(x→−1) (x² − 3x − 1). → 3Prompt: Polynomials are continuous: you can substitute directly.
- For y = x², chords from x = −2 to x = −2 + h have gradients: h = 0.1: −3.9 h = 0.01: −3.99 h = 0.001: −3.999 What is the gradient of the curve at x = −2? → -4Prompt: As h shrinks, the chord gets closer to the tangent.
Hook · 5 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Differentiate from first principles. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: For f(x) = x² − 4x, simplify (f(x + h) − f(x))/h.
- Step 1: (2xh + h² − 4h)/h
- Step 2: 2x + h − 4
- Answer: 2x + h − 4
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- For f(x) = x² + 4x, simplify (f(x + h) − f(x))/h. (2xh + h² / 2x + h + 4 / 2x + 4 / x² + h + 4) → 2x + h + 4Prompt: Expand f(x + h) = (x + h)² + b(x + h).
- Using first principles, f(x) = 2x² + 3x gives (f(x + h) − f(x))/h = 4x + 2h + 3. What is f′(x)? → 4x+3Prompt: f′(x) is the limit as h → 0.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Using first principles, f(x) = 3x² − 3x gives (f(x + h) − f(x))/h = 6x + 3h − 3. What is f′(x)? → 6x-3Prompt: f′(x) is the limit as h → 0.
- For f(x) = x² − x, simplify (f(x + h) − f(x))/h. (2x + h − 1 / 2x − 1 / 2x + h² − 1 / x² + h − 1) → 2x + h − 1Prompt: Expand f(x + h) = (x + h)² + b(x + h).
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Estimate the gradient of a curve with chords.
- Use concrete materials: graphing tool or graph paper.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Differentiate axⁿ with the power rule.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder