Differentiation in 9709: Stationary Points, Rates of Change and Tangents

Differentiation is topic 1.7 of Pure Mathematics 1 in the Cambridge International AS & A Level Mathematics 9709 syllabus (2026–2027). It is one of the most reliable sources of marks on Paper 1. The questions follow recognisable patterns, and each pattern rewards a clear, well-set-out method. This guide covers what the syllabus requires, then works through seven original exam-style examples.

What the syllabus requires

Under topic 1.7, candidates should be able to:

• understand the gradient of a curve as the limit of gradients of chords (only an informal idea of a limit is expected), and use the notations f′(x), f″(x), dy/dx and d²y/dx²
• differentiate xⁿ for any rational n, with constant multiples, sums and differences, and composite functions using the chain rule
• apply differentiation to gradients, tangents and normals, increasing and decreasing functions, and rates of change (including connected rates of change)
• locate stationary points and determine their nature, including by the second derivative, and use them in sketching graphs

Two boundaries are worth knowing. Formal differentiation from first principles is not required, and points of inflexion are not included at this stage. The product and quotient rules, and the differentiation of eˣ, ln x and trigonometric functions, belong to Pure Mathematics 2 and 3 (Papers 2 and 3).

The core method in one line

Write the function in index form, differentiate term by term (using the chain rule for brackets), then use the derivative for whatever the question asks: a gradient, an equation, a stationary point or a rate.

Example 1: stationary points and their nature

Question. The curve has equation y = x³ − 6x² + 9x + 2. Find the coordinates of the stationary points and determine their nature.

Solution.

• dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3)
• Setting dy/dx = 0 gives x = 1 or x = 3.
• At x = 1: y = 1 − 6 + 9 + 2 = 6. At x = 3: y = 27 − 54 + 27 + 2 = 2.
• d²y/dx² = 6x − 12. At x = 1 it is −6 < 0, so (1, 6) is a maximum. At x = 3 it is 6 > 0, so (3, 2) is a minimum.

Always state the conclusion in words. A correct second derivative with no stated conclusion can lose the final mark.

Example 2: tangent and normal

Question. Find the equations of the tangent and the normal to y = 2x² − 3x + 1 at the point where x = 2.

Solution.

• At x = 2: y = 8 − 6 + 1 = 3.
• dy/dx = 4x − 3, which is 5 at x = 2.
• Tangent: y − 3 = 5(x − 2), so y = 5x − 7.
• The normal gradient is −1/5: y − 3 = −(1/5)(x − 2), so x + 5y = 17.

Check: (2, 3) satisfies both equations.

Example 3: the chain rule

Question. Find dy/dx when (a) y = (3x − 1)⁵, (b) y = √(4x + 1), and evaluate (b) at x = 2.

Solution.

• (a) dy/dx = 5(3x − 1)⁴ × 3 = 15(3x − 1)⁴
• (b) y = (4x + 1)^(1/2), so dy/dx = ½(4x + 1)^(−1/2) × 4 = 2/√(4x + 1). At x = 2: 2/√9 = 2/3.

The most common error is forgetting to multiply by the derivative of the inside bracket.

Example 4: connected rates of change

Question. The radius of a sphere increases at 0.2 cm s⁻¹. Find the rate of increase of the volume when the radius is 5 cm. (The volume of a sphere is V = (4/3)πr³.)

Solution.

• dV/dr = 4πr², which is 100π at r = 5.
• dV/dt = (dV/dr) × (dr/dt) = 100π × 0.2 = 20π ≈ 62.8 cm³ s⁻¹.

Write the chain of rates explicitly. It earns method marks even if the arithmetic slips.

Example 5: increasing and decreasing functions

Question. Show that f(x) = x³ + 3x² + 3x − 4 is an increasing function.

Solution.

• f′(x) = 3x² + 6x + 3 = 3(x + 1)².
• Since (x + 1)² ≥ 0 for all x, f′(x) ≥ 0, with equality only at x = −1. Hence f is an increasing function.

Completing the square, or factorising as here, is the standard way to show that a derivative never changes sign.

Example 6: optimisation

Question. A farmer has 60 m of fencing to enclose a rectangular area against a straight wall. The wall forms one side and needs no fencing. Find the maximum area.

Solution.

• Let each side perpendicular to the wall be x m. The side parallel to the wall is then (60 − 2x) m.
• A = x(60 − 2x) = 60x − 2x²
• dA/dx = 60 − 4x = 0, so x = 15.
• d²A/dx² = −4 < 0, so this is a maximum.
• Maximum area = 15 × 30 = 450 m²

Example 7: negative indices

Question. Find the stationary point of y = x + 4/x for x > 0 and determine its nature.

Solution.

• y = x + 4x⁻¹, so dy/dx = 1 − 4x⁻² = 0, giving x² = 4 and x = 2 (since x > 0).
• At x = 2: y = 2 + 2 = 4.
• d²y/dx² = 8x⁻³, which is 1 > 0 at x = 2, so (2, 4) is a minimum.

Mistakes that cost marks

1. Differentiating 4/x as 4 instead of rewriting it as 4x⁻¹ first.
2. Missing the inner derivative in the chain rule.
3. Using the tangent gradient for the normal (the normal gradient is −1/m).
4. Finding x at a stationary point but not y.
5. Not stating "maximum" or "minimum" after the second-derivative test.
6. Rounding mid-question. 9709 expects non-exact answers to 3 significant figures unless the question says otherwise.

How to practise

• Generate topic sets from the 9709 topical questions bank, then attempt full Paper 1s from the 9709 past papers library.
• Read 9709 Paper 1 tips, which covers presentation and the errors examiners see most often.
• Continue to the reverse process with 9709 P3 integration: 10 exam-style questions.
• For where differentiation sits in the course, see the 9709 Syllabus 2026–2027 guide.

Frequently asked questions

Are points of inflexion on Paper 1? No. The 9709 syllabus states that knowledge of points of inflexion is not included in topic 1.7.

Do I have to use the second derivative to classify stationary points? Not always. The syllabus includes the second-derivative test, but alternatives may be used where the question does not specify a method, such as checking the sign of the gradient either side.

Is differentiation from first principles examined? Formal use of first principles is not required. An informal understanding of the gradient as the limit of chord gradients is expected.

Where do the product and quotient rules appear? In Pure Mathematics 2 and Pure Mathematics 3 (Papers 2 and 3), not in Paper 1.

Sources: Cambridge International AS & A Level Mathematics 9709 syllabus for examination in 2026–2027: topic 1.7 (Differentiation) with its notes, the differentiation sections of Pure Mathematics 2 and 3, and Degrees of accuracy. All examples are original.

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