amazon.co.ukDifferentiation
A-Level Mathematics revision: Differentiation. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Power rule: d/dx(xⁿ) = nxⁿ⁻¹ for all real n
Key Fact: Sum/difference: d/dx(f+/-g) = f' +/- g'
Key Fact: Product rule: d/dx(fg) = f'g + fg'
Key Fact: Quotient rule: d/dx(f/g) = (f'g - fg')/g^2
Key Fact: Chain rule: d/dx(f(g(x))) = f'(g(x))·g'(x)
Key Fact: Stationary points: f'(x) = 0. Max: f'' < 0; Min: f'' > 0; Inflection: f'' = 0 with sign change
Key Fact: Tangent: y - y₁ = f'(x₁)(x - x₁); Normal gradient = -1/f'(x₁)
Key Fact: Optimisation: find critical points, check endpoints, justify global max/min
Key Fact: Implicit: differentiate term-by-term w.r.t x, solve for dy/dx
Key Fact: Connected rates: dy/dt = dy/dx · dx/dt
🎯 Learning Objectives
- Differentiate polynomials, rational and negative powers from first principles
- Apply sum/difference, product, quotient and chain rules
- Find gradients of tangents and normals to curves
- Find and classify stationary points (maxima, minima, inflection)
- Apply second derivative test for nature of stationary points
- Solve optimisation problems (max/min area, volume, cost)
- Use implicit differentiation for curves not in y = f(x) form
- Understand connected rates of change
💡 Worked Example
Exam-Style Question
Question: Find the stationary points of y = x^3 - 3x^2 - 9x + 5 and determine their nature
Model Answer:
y' = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1) = 0 -> x = 3 or x = -1. y'' = 6x - 6. At x = 3: y'' = 12 > 0 -> minimum at (3, -22). At x = -1: y'' = -12 < 0 -> maximum at (-1, 10)
❓ Practice Questions
Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.
Questions:
- Differentiate: xsqrt(x+2) using product rule✗ answer coming soon
- Find the equation of the normal to y = x^2 - 4x + 3 at x = 2✗ answer coming soon
- A rectangular box with square base has volume 500 cm^3. Find minimum surface area✗ answer coming soon
- For x^2 + y^2 = 25, find dy/dx when x = 3 (implicit)✗ answer coming soon
- Radius of a sphere increases at 0.5 cm/s. Rate of volume increase when r = 10 cm?✗ answer coming soon
🎬 Video Resources
- Differentiation - Overview -- YouTube Search
- Differentiation - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Differentiation -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Differentiation -- Free resources, slides, and interactive questions
- BBC Bitesize - Differentiation -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of differentiation with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Find the stationary points of y = x^3 - 3x^2 - 9x + 5 and determine their nature. Solution: y' = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1) = 0 -> x = 3 or x = -1. y'' = 6x - 6. At x = 3: y'' = 12 > 0 -> minimum at (3, -22). At x = -1: y'' = -12 < 0 -> maximum at (-1, 10)
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Differentiate: xsqrt(x+2) using product rule
- Find the equation of the normal to y = x^2 - 4x + 3 at x = 2
- A rectangular box with square base has volume 500 cm^3. Find minimum surface area
- For x^2 + y^2 = 25, find dy/dx when x = 3 (implicit)
- Radius of a sphere increases at 0.5 cm/s. Rate of volume increase when r = 10 cm?
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.