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Differentiation practice problems A level Maths, Topic 5

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

Differentiating powers, ex and ln x, the chain, product and quotient rules, tangents, stationary points and connected rates of change. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Exam question practice
  5. Common mistakes
  6. Keep going

Key equations for differentiation

d/dx(xn) = nxn−1power rule
d/dx(ekx) = kekx, d/dx(ln x) = 1/xexponentials and logs
dy/dx = dy/du × du/dxchain rule
d/dx(uv) = u′v + uv′product rule

A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Find the value of d/dx(4x3) at x = 2.

Hint

Differentiate, then substitute.

Worked solution
  1. d/dx(4x3) = 12x2.
  2. At x = 2: 12 × 4 = 48.
Q2Warm-up
Find the gradient of y = x2 − 3x at x = 5.

Hint

dy/dx = 2x − 3.

Worked solution
  1. dy/dx = 2(5) − 3 = 7.
Q3Warm-up
What is d/dx(e3x)?

Hint

Use the chain rule with u = 3x.

Worked solution
  1. d/dx(e3x) = 3e3x: B.

Exam-style questions

Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Find the x-coordinate of the stationary point of y = x2 − 6x + 2.

Hint

Set dy/dx = 0.

Worked solution
  1. dy/dx = 2x − 6 = 0.
  2. x = 3.
Q5Exam-style
y = (2x + 1)4. Find dy/dx at x = 0.

Hint

Chain rule: 4(2x + 1)3 × 2.

Worked solution
  1. dy/dx = 8(2x + 1)3.
  2. At x = 0: 8.
Q6Exam-styleYear 2
y = xex. Find dy/dx at x = 1, to 3 significant figures.

Hint

Product rule with u = x and v = ex.

Worked solution
  1. dy/dx = ex + xex.
  2. At x = 1: 2e = 5.44.
Q7Exam-style
Find the gradient of the tangent to y = ln x at x = 4.

Hint

d/dx(ln x) = 1/x.

Worked solution
  1. dy/dx = 1/4 = 0.25.
Q8ChallengeYear 2
The area of a circle increases at 10 cm²/s. Find the rate of increase of the radius when r = 5 cm, to 3 significant figures.
cm/s

Hint

dA/dt = dA/dr × dr/dt, with A = πr2.

Worked solution
  1. dA/dr = 2πr = 10π at r = 5.
  2. dr/dt = 10/(10π) = 1/π = 0.318 cm/s.

Exam question practice

One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.

Exam-style question style · 9 points · about 9 minutes

An open box is made from a 12 cm by 12 cm square of card by cutting a square of side x cm from each corner and folding up the sides.

  1. (a) Show that the volume is V = x(12 − 2x)2.
  2. (b) Find the value of x that gives the maximum volume.
  3. (c) Find the maximum volume and show that it is a maximum.
  4. (d) Explain why x must be less than 6.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

The base is (12 − 2x) by (12 − 2x) and the height is x.

Part (b)

V = 144x − 48x2 + 4x3. dV/dx = 144 − 96x + 12x2 = 12(x − 2)(x − 6) = 0. x = 6 gives no box, so x = 2.

Part (c)

V = 2 × 82 = 128 cm3. d2V/dx2 = 24x − 96 = −48 < 0 at x = 2, so it is a maximum.

Part (d)

The side 12 − 2x must be positive.

Your mark: 0 of 9

Common mistakes in differentiation

Keep going

Revise the ideas behind these questions:

Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units