Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Differentiate axⁿ with the power rule
Calculus
CCSS HSF.IF.B.6England: A-level Pure Mathematics
Objective and success criteria
Students will: Differentiate axⁿ with the power rule.
- 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: Differentiate from first principles; Laws of exponents (indices)
Leads to: Differentiate polynomials term by term; Differentiate negative and fractional powers; Integrate powers of x (indefinite integrals)
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 (Differentiate from first principles; Laws of exponents (indices)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Simplify (a³)⁵ and write it as a to a single power. What is the power? → 15Prompt: A power of a power: multiply the exponents.
- For f(x) = x² + 3x, simplify (f(x + h) − f(x))/h. (x² + h + 3 / 2x + h + 3 / 2x + 3 / 2xh + h²) → 2x + h + 3Prompt: Expand f(x + h) = (x + h)² + b(x + h).
- Using first principles, f(x) = x² − 4x gives (f(x + h) − f(x))/h = 2x + h − 4. What is f′(x)? → 2x-4Prompt: f′(x) is the limit as h → 0.
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 axⁿ with the power rule. 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: Differentiate y = −4x².
- Step 1: dy/dx = -4 × 2x
- Step 2: = −8x
- Answer: -8*x^1
- 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.
- Differentiate y = −4x³. → -12*x^2Prompt: Multiply by the power, then reduce the power by 1.
- Kwame differentiated 6x⁵ and wrote 30x⁶. What is the correct derivative? (30x⁴ / 6x⁴ / 24x⁴ / 30x⁶) → 30x⁴Prompt: The power goes down by one, not up.
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.
- Ben differentiated 4x³ and wrote 12x⁴. What is the correct derivative? (12x² / 8x² / 12x⁴ / 4x²) → 12x²Prompt: The power goes down by one, not up.
- Differentiate y = −3x⁶. → -18*x^5Prompt: Multiply by the power, then reduce the power by 1.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Differentiate from first principles.
- 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 polynomials term by term; Differentiate negative and fractional powers; Integrate powers of x (indefinite integrals).
- 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