Free Grade 3 Reverse Differentiation Lessons
Free Grade 3 lessons for Reverse Differentiation: 12 real questions with a full answer key and worked solutions. Differentiation finds the rate of change (the gradient) of a curve. For axⁿ, the derivative is naxⁿ⁻¹.
Free
Foundation
30 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- RememberIdentify reverse differentiation
Identify reverse differentiation accurately in a familiar context.
Assessed by: Exit ticket of four short items
- UnderstandExplain reverse differentiation
Explain reverse differentiation accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate reverse differentiation
Calculate reverse differentiation accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare reverse differentiation
Compare reverse differentiation accurately in a familiar context.
Assessed by: Exit ticket of four short items
Prerequisites
Close these gaps first — each has its own free lesson, worksheet and quiz.
How Reverse Differentiation works
Differentiation finds the rate of change (the gradient) of a curve. For axⁿ, the derivative is naxⁿ⁻¹.
12 practice questions
Easy to hard, 24 marks in total. Try them first, then open the answers.
- 1.Differentiate y = 3x^4 + 1x.[1]
- 2.Differentiate y = 5x^2 + 4x.[1]
- 3.Differentiate y = 1x^2 + 6x.[1]
- 4.Differentiate y = 2x^3 + 7x.[1]
- 5.Differentiate y = 4x^2 + 6x.[2]
- 6.Differentiate y = 6x^4 + 3x.[2]
- 7.Differentiate y = 5x^2 + 6x.[2]
- 8.Differentiate y = 3x^3 + 6x.[2]
- 9.Find the gradient of y = 1x^2 + 4x at x = 1.[3]
- 10.Find the gradient of y = 6x^2 + 3x at x = 3.[3]
- 11.Find the gradient of y = 1x^2 + 9x at x = 2.[3]
- 12.Find the gradient of y = 1x^3 + 3x at x = 2.[3]
Show answers and working
1. dy/dx = 12x^3 + 1
Multiply by the power, then reduce the power by 1. 3x^4 → 12x^3; 1x → 1.
2. dy/dx = 10x^1 + 4
Multiply by the power, then reduce the power by 1. 5x^2 → 10x^1; 4x → 4.
3. dy/dx = 2x^1 + 6
Multiply by the power, then reduce the power by 1. 1x^2 → 2x^1; 6x → 6.
4. dy/dx = 6x^2 + 7
Multiply by the power, then reduce the power by 1. 2x^3 → 6x^2; 7x → 7.
5. dy/dx = 8x^1 + 6
Multiply by the power, then reduce the power by 1. 4x^2 → 8x^1; 6x → 6.
6. dy/dx = 24x^3 + 3
Multiply by the power, then reduce the power by 1. 6x^4 → 24x^3; 3x → 3.
7. dy/dx = 10x^1 + 6
Multiply by the power, then reduce the power by 1. 5x^2 → 10x^1; 6x → 6.
8. dy/dx = 9x^2 + 6
Multiply by the power, then reduce the power by 1. 3x^3 → 9x^2; 6x → 6.
9. 6
dy/dx = 2x^1 + 4. At x = 1: 2 × 1 + 4 = 6.
10. 39
dy/dx = 12x^1 + 3. At x = 3: 12 × 3 + 3 = 39.
11. 13
dy/dx = 2x^1 + 9. At x = 2: 2 × 2 + 9 = 13.
12. 15
dy/dx = 3x^2 + 3. At x = 2: 3 × 4 + 3 = 15.
Worked examples
Differentiate y = 2x^4 + 1x.
- Multiply by the power, then reduce the power by 1.
- 2x^4 → 8x^3; 1x → 1.
- Answer: dy/dx = 8x^3 + 1
Differentiate y = 3x^2 + 4x.
- Multiply by the power, then reduce the power by 1.
- 3x^2 → 6x^1; 4x → 4.
- Answer: dy/dx = 6x^1 + 4
Find the gradient of y = 5x^5 + 5x at x = 3.
- dy/dx = 25x^4 + 5.
- At x = 3: 25 × 81 + 5 = 2030.
- Answer: 2030
Interactive practice
Free, no account required — answer on screen or print it.
Common mistakes
What learners typically get wrong, and how to fix it.
Forgets that a constant differentiates to 0.
Correction: Constants have zero gradient.
Reduces the power but forgets to multiply by it.
Correction: Multiply by the power, then reduce it by 1.
Teacher tips
- · Link to gradients of straight lines first.
Parent tips
- · Ask how speed relates to distance over time.
Real-life applications
- · Speed from distance, maximising profit.
Assessment objectives
How mastery is evidenced, and which standards it maps to.
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Frequently asked
- Is this Grade 3 Reverse Differentiation lesson really free?
- Yes. Every lessons on FreeMathWorksheet is free to use, print and share — no account, no paywall and no watermark.
- What should a learner already know before this?
- Start with Differentiation. Each one has its own free lesson, worksheet and quiz.
- How is the lesson sequenced?
- Recognition first, then understanding, application, reasoning and finally mastery — the same ladder used across every free resource on the platform.
- What comes next after Reverse Differentiation?
- Move on to Definite Integrals, Area Under a Curve, Differentiation, which build directly on this idea.
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