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 (y⁹)⁵ and write it as y to a single power. What is the power? → 45Prompt: A power of a power: multiply the exponents.
- For f(x) = x² + 4x, simplify (f(x + h) − f(x))/h. (x² + h + 4 / 2xh + h² / 2x + h + 4 / 2x + 4) → 2x + h + 4Prompt: Expand f(x + h) = (x + h)² + b(x + h).
- Using first principles, f(x) = 5x² + 3x gives (f(x + h) − f(x))/h = 10x + 5h + 3. What is f′(x)? → 10x+3Prompt: 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 = −9x⁵. → -45*x^4Prompt: Multiply by the power, then reduce the power by 1.
- Ana differentiated 7x⁵ and wrote 35x⁶. What is the correct derivative? (7x⁴ / 35x⁴ / 35x⁶ / 28x⁴) → 35x⁴Prompt: 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.
- Differentiate y = −x⁶. → -6*x^5Prompt: Multiply by the power, then reduce the power by 1.
- Jamal differentiated 9x² and wrote 18x³. What is the correct derivative? (11x / 18x / 18x³ / 9x) → 18xPrompt: The power goes down by one, not up.
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