Mathematics 9709 · syllabus 1.7, 2.4, 3.4
Cambridge A Level Mathematics 9709 — Differentiation topical past papers
Gradients, tangents, rates of change and stationary points, from the chain rule in Paper 1 to product, quotient, implicit and parametric work later.
- Questions
- 367
- Years
- 2009–2026
- With mark scheme
- 99%
- Questions 2022–2026
- 114
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Differentiation covers
Paper 1 (section 1.7) treats the derivative as the gradient of a curve, the limit of the gradients of chords. You differentiate xⁿ for any rational n, sums and multiples, and composite functions with the chain rule. You then use derivatives for tangents and normals, increasing and decreasing functions, connected rates of change, and stationary points, using the second derivative to tell a maximum from a minimum. Points of inflexion are not required.
Papers 2 and 3 (sections 2.4 and 3.4) add the derivatives of eˣ, ln x, sin x, cos x and tan x, the product and quotient rules, and the first derivative of a curve defined parametrically or implicitly, used for tangents and normals. Paper 3 also includes the derivative of tan⁻¹ x. The Paper 1 applications carry over, so a later question may ask for the stationary points of x²e⁻ˣ or the tangent to an implicitly defined curve.
Differentiation starts most of the calculus in the course. Integration reverses it, kinematics in Paper 4 uses it to find velocity and acceleration from displacement, and the differential equations of Paper 3 are built from rates of change. It also leans on earlier work: tangents and normals are straight lines from coordinate geometry, the stationary points of a cubic come from solving a quadratic, and finding where a gradient takes a given value can need the trigonometric or logarithmic equations of Papers 2 and 3.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.7 Differentiation
- 2.4 Differentiation
- 3.4 Differentiation
What you should be able to do
- Differentiate xⁿ for rational n, and composite functions using the chain rule
- Find the equations of tangents and normals at a given point
- Find stationary points and determine their nature with the second derivative
- Solve connected rates of change problems with the chain rule
- Differentiate eˣ, ln x and trigonometric functions, with products and quotients (Papers 2 and 3)
- Find dy/dx for curves defined parametrically or implicitly (Papers 2 and 3)
Key ideas
- d/dx(xⁿ) = nxⁿ⁻¹; chain rule dy/dx = (dy/du) × (du/dx)
- At a stationary point dy/dx = 0; d²y/dx² > 0 there gives a minimum and d²y/dx² < 0 a maximum
- Product rule d(uv)/dx = u dv/dx + v du/dx; quotient rule d(u/v)/dx = (v du/dx − u dv/dx)/v²
- d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, with x in radians
- Parametric: dy/dx = (dy/dt) ÷ (dx/dt); in Paper 3, d/dx(tan⁻¹x) = 1/(1 + x²)
How Differentiation has been examined
From 2022 to 2026, Differentiation was on Paper 1 · Pure Mathematics 1 in 11 of 11 exam series, worth about 20% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 60 | 20% | 11 of 11 |
| Paper 3 · Pure Mathematics 3 | 54 | 17% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 8.
Filed in Momentum under: Differentiation.
A worked example
A curve has equation y = x³ − 6x² + 9x + 2. Find the coordinates of its stationary points and determine their nature.
- Differentiate: dy/dx = 3x² − 12x + 9.
- At stationary points dy/dx = 0: 3(x² − 4x + 3) = 0, so 3(x − 1)(x − 3) = 0, giving x = 1 or x = 3.
- Substitute into the curve: at x = 1, y = 1 − 6 + 9 + 2 = 6; at x = 3, y = 27 − 54 + 27 + 2 = 2.
- Second derivative: d²y/dx² = 6x − 12, which is −6 at x = 1 and 6 at x = 3.
- A negative second derivative means a maximum and a positive one a minimum.
Answer: (1, 6) is a maximum point and (3, 2) is a minimum point
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Leaving out the chain rule's inner derivative: d/dx (3x − 1)⁵ is 15(3x − 1)⁴, not 5(3x − 1)⁴.
- Using the tangent's gradient in the equation of the normal: the normal's gradient is −1/m.
- Differentiating y² as 2y in implicit differentiation instead of 2y dy/dx, which loses the dy/dx terms you need to collect.
- Differentiating trigonometric functions with x in degrees: the derivative of sin x is cos x only when x is in radians.
- Stating the nature of a stationary point without evidence: the value or sign of the second derivative, or of dy/dx either side, has to be shown.
Exam technique
- Rewrite roots and fractions as powers before differentiating, so √x becomes x^(1/2) and 4/x² becomes 4x⁻².
- For connected rates of change, write the chain of rates with units, such as dA/dt = (dA/dr) × (dr/dt), and substitute values only after differentiating.
- When asked for the nature of a stationary point, show the second derivative's value or sign and state the conclusion in words.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.