IGCSE Math RevisionWorked example 1: Power rule
Differentiate \(y=5x^4-3x^2+2x-7\).
Solution
- Multiply by the power, then reduce the power by one.
- \(\dfrac{dy}{dx}=20x^3-6x+2\).
Answer: \(20x^3-6x+2\)
Tip: A constant differentiates to 0.
Edexcel IGCSE Maths (4MA1) · Unit 8: Calculus
IGCSE Maths Differentiation Questions and Answers
Differentiation questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 5 worked examples first, then cover each solution and redo it yourself.
- 5 worked examples
- Higher tier · calculator allowed
- Free, no sign-in
What this topic covers
- Differentiation 1: Power Rule: Differentiating polynomial terms with the power rule.
- Differentiation 2: Gradients of Curves: Finding gradient at a specific x-value.
- Differentiation 3: Tangents & Normals: Equations of tangents and normals at a point: gradient m from dy/dx, normal gradient −1/m, then y − y₁ = m(x − x₁).
- Differentiation 4: Stationary Points: Finding stationary points from dy/dx = 0.
- Differentiation 5: Turning Points Optimization: Classifying max/min and optimisation problems.
Differentiation worked examples
Worked example 2: Gradient of a curve
Find the gradient of \(y=x^2-5x+2\) at \(x=4\).
Solution
- \(\dfrac{dy}{dx}=2x-5\).
- At \(x=4\): \(8-5=3\).
Answer: 3
Tip: Substitute into dy/dx, not into y.
Worked example 3: The tangent and the normal at a point
The curve \(y=x^3-2x\) passes through the point \((2,\ 4)\). Find (a) the equation of the tangent to the curve at this point, (b) the equation of the normal to the curve at this point.
Solution
- \(\dfrac{dy}{dx}=3x^2-2\); at \(x=2\) the gradient of the tangent is 10.
- (a) Tangent: \(y-4=10(x-2)\), so \(y=10x-16\).
- (b) The normal is perpendicular to the tangent, so its gradient is \(-\tfrac{1}{10}\).
- Normal: \(y-4=-\tfrac{1}{10}(x-2)\); multiply by 10: \(10y-40=-x+2\), so \(x+10y=42\).
Answer: (a) \(y=10x-16\) (b) \(x+10y=42\)
Tip: The tangent has the same gradient as the curve at that point; the normal is perpendicular to it, so its gradient is \(-\tfrac{1}{m}\). In 4MA1 the gradient may come from \(\dfrac{dy}{dx}\) or from a tangent drawn on a graph.
Worked example 4: Stationary points
Find the stationary points of \(y=2x^3-3x^2-12x\).
Solution
- \(\dfrac{dy}{dx}=6x^2-6x-12=0\), so \(x^2-x-2=0\).
- \((x-2)(x+1)=0\): \(x=2\) or \(x=-1\).
- \((2,\ -20)\) and \((-1,\ 7)\): for a positive cubic, \((-1,7)\) is the maximum and \((2,-20)\) the minimum.
Answer: Maximum \((-1,\ 7)\), minimum \((2,\ -20)\)
Tip: Find y by substituting into the original equation.
Worked example 5: Optimisation
A farmer has 60 m of fencing to make three sides of a rectangular pen against a wall. Find the maximum area.
Solution
- Two sides of \(x\) and one of \(60-2x\): \(A=60x-2x^2\).
- \(\dfrac{dA}{dx}=60-4x=0\), so \(x=15\).
- \(A=15\times30=450\) m².
Answer: 450 m²
Tip: Write the area in terms of one variable before differentiating.
Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.
Revise differentiation for the exam
- Unit 8 practice: Calculus in the student hub (more questions with mark schemes)
- IGCSE Maths revision notes for this topic and the IGCSE Maths formula sheet
- Edexcel IGCSE Maths 4MA1 past papers to practise under exam conditions
- IGCSE Maths 4MA1 examiner report summary: where students lose marks
Questions students ask
Which Edexcel IGCSE Maths papers test differentiation?
Either Higher tier paper, 1H or 2H, can test differentiation: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.
Where can I practise more differentiation questions?
In the IGCSE student hub: Unit 8 (Calculus) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.