IGCSE Math RevisionQuestion 1
\(y = 5x^3\). Find \(\frac{dy}{dx}\).
Show the worked answer
- Multiply by the power and reduce the power by \(1\): \(5 \times 3x^2\).
Answer: \(\frac{dy}{dx} = 15x^2\)
Cambridge IGCSE Maths (0580) · Topic 2: Algebra and graphs · 2.12
Cambridge IGCSE Maths 0580 Differentiation Questions
Differentiation questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.
- 8 questions
- 8 Extended
- 7 non-calculator
- Free, no sign-in
What the syllabus asks
- 2.12 Differentiation Extended only
Extended: Estimate gradients of curves by drawing tangents; Differentiate axⁿ (a rational, n a positive integer or zero) and simple sums of up to three such terms, using dy/dx notation; Gradients and stationary (turning) points; decide between maxima and minima by any method (no points of inflection).
Summarised in our own words: check the exact wording in the official 0580 syllabus.
Differentiation questions: Extended
Extended content, for Papers 2 and 4.
Question 2
\(y = 4x^2 - 3x + 7\). Find \(\frac{dy}{dx}\).
Show the worked answer
- \(4x^2 \to 8x\), \(-3x \to -3\), and the constant \(7 \to 0\).
Answer: \(\frac{dy}{dx} = 8x - 3\)
Question 3
Find the gradient of the curve \(y = x^3 - 2x\) at the point where \(x = 2\).
Show the worked answer
- \(\frac{dy}{dx} = 3x^2 - 2\)
- At \(x = 2\): \(12 - 2 = 10\).
Answer: \(10\)
Question 4
The gradient of the curve \(y = x^2 - 6x + 1\) is \(4\) at the point \(P\).
Find the \(x\)-coordinate of \(P\).
Show the worked answer
- \(\frac{dy}{dx} = 2x - 6\)
- \(2x - 6 = 4\), so \(x = 5\).
Answer: \(x = 5\)
Question 5
Find the coordinates of the turning point of \(y = x^2 - 8x + 3\), and say whether it is a maximum or a minimum.
Show the worked answer
- \(\frac{dy}{dx} = 2x - 8 = 0\), so \(x = 4\).
- \(y = 16 - 32 + 3 = -13\).
- The \(x^2\) term is positive, so the curve is U-shaped: a minimum.
Answer: \((4, -13)\), minimum
Question 6
Find the coordinates of the two turning points of \(y = x^3 - 3x^2 - 9x + 2\) and determine which is a maximum.
Show the worked answer
- \(\frac{dy}{dx} = 3x^2 - 6x - 9 = 3(x - 3)(x + 1)\), so \(x = 3\) or \(x = -1\).
- \(x = -1\): \(y = -1 - 3 + 9 + 2 = 7\). \(x = 3\): \(y = 27 - 27 - 27 + 2 = -25\).
- A positive cubic rises, falls, then rises, so the first turning point is the maximum.
Answer: \((-1, 7)\) maximum, \((3, -25)\) minimum
Question 7
Find the equation of the tangent to \(y = x^2 + 3x\) at the point where \(x = 1\).
Show the worked answer
- At \(x = 1\), \(y = 4\). \(\frac{dy}{dx} = 2x + 3 = 5\).
- \(y - 4 = 5(x - 1)\), so \(y = 5x - 1\).
Answer: \(y = 5x - 1\)
Question 8
A rectangle has a perimeter of \(40\) cm and a width of \(x\) cm, so its area is \(A = x(20 - x)\) cm².
Use differentiation to find the greatest possible area.
Show the worked answer
- \(A = 20x - x^2\), so \(\frac{dA}{dx} = 20 - 2x\).
- \(20 - 2x = 0\) gives \(x = 10\). \(A = 10 \times 10 = 100\).
Answer: \(100\) cm²
Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).
Keep going
- Algebra and graphs practice for 0580: more questions, each one marked, tagged Core or Extended
- My mistakes, then a Skill Builder on the skills you missed
- Common mistakes in 0580: what the Cambridge examiner reports say
- Algebra and graphs revision notes and the 0580 formula sheet
- Cambridge IGCSE Maths 0580 past papers, when you are ready for a full paper
Questions students ask
Which Cambridge 0580 papers can test differentiation?
Differentiation is Extended content (syllabus 2.12), so it is tested only on the Extended papers, Paper 2 (non-calculator) and Paper 4 (calculator). Core candidates do not need it.
Are these Cambridge past-paper questions?
No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.
Where can I practise more differentiation questions?
In the 0580 practice for topic 2 (Algebra and graphs), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.