A Level Maths: Pure 3 chapter 4: Differentiation MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

32 exam-style questions on Differentiation, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

32 questions · 1 topics · 10 per test

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What chapter 4 covers

Ten Differentiation questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. Differentiate x2e3x with respect to x.

  1. A
    6xe3x
  2. B
    xe3x(x + 2)
  3. C
    3x2e3x
  4. D
    xe3x(3x + 2)
Show answer

Answer: D. Product rule: 2x × e3x + x2 × 3e3x = xe3x(2 + 3x). The derivative of a product is not the product of the derivatives.

2. How many stationary points does the curve y = sin x + cos 2x have for 0 < x < π?

  1. A
    2
  2. B
    3
  3. C
    1
  4. D
    4
Show answer

Answer: B. dy/dx = cos x − 2 sin 2x = cos x − 4 sin x cos x = cos x(1 − 4 sin x). cos x = 0 gives x = π/2, and sin x = ¼ gives two values between 0 and π, so there are 3. Dividing by cos x loses x = π/2.

3. Differentiate (ln x)/x with respect to x.

  1. A
    (1 − ln x)/x2
  2. B
    (ln x − 1)/x2
  3. C
    (1 + ln x)/x2
  4. D
    1/x
Show answer

Answer: A. Quotient rule: [x × (1/x) − ln x × 1]/x2 = (1 − ln x)/x2. The order in the numerator matters (v du/dx − u dv/dx), so swapping it gives the wrong sign.

4. Differentiate tan 3x with respect to x.

  1. A
    sec2 3x
  2. B
    3 sec 3x tan 3x
  3. C
    3 sec2 3x
  4. D
    −3 cosec2 3x
Show answer

Answer: C. The derivative of tan x is sec2 x, and the chain rule multiplies by 3. 3 sec 3x tan 3x is the derivative of sec 3x.

5. Differentiate e3x − 1 with respect to x.

  1. A
    e3x − 1
  2. B
    (3x − 1)e3x − 2
  3. C
    ⅓e3x − 1
  4. D
    3e3x − 1
Show answer

Answer: D. By the chain rule the derivative of ef(x) is f′(x)ef(x), so the answer is 3e3x − 1. The power rule does not apply to an exponential.

6. Differentiate (2x + 3)/(x − 1) with respect to x.

  1. A
    5/(x − 1)2
  2. B
    1/(x − 1)2
  3. C
    −5/(x − 1)2
  4. D
    2
Show answer

Answer: C. [2(x − 1) − (2x + 3)(1)]/(x − 1)2 = (2x − 2 − 2x − 3)/(x − 1)2 = −5/(x − 1)2. Removing the bracket −(2x + 3) needs −3, not +3.

7. Find dy/dx for the curve 3x2 + y2 = 7, in terms of x and y.

  1. A
    3x/y
  2. B
    −3x
  3. C
    −3x/y
  4. D
    −y/(3x)
Show answer

Answer: C. Differentiate implicitly: 6x + 2y dy/dx = 0, so dy/dx = −6x/(2y) = −3x/y. The derivative of y2 with respect to x is 2y dy/dx, not 2 dy/dx.

8. Differentiate e−x cos x with respect to x.

  1. A
    −e−x(cos x + sin x)
  2. B
    e−x(sin x − cos x)
  3. C
    e−x(cos x − sin x)
  4. D
    −e−x sin x
Show answer

Answer: A. Product rule: (−e−x)cos x + e−x(−sin x) = −e−x(cos x + sin x). Both factors bring a minus sign when differentiated.

9. Find the gradient of the curve y = x sin 2x at the point where x = π/6.

  1. A
    √3/2 + π/6
  2. B
    √3/2 + π/12
  3. C
    √3/2 − π/6
  4. D
    π/6
Show answer

Answer: A. dy/dx = sin 2x + 2x cos 2x. At x = π/6: sin(π/3) + (π/3)cos(π/3) = √3/2 + π/6. Leaving out the chain-rule factor 2 gives π/12 for the second term.

10. A curve has parametric equations x = 2 cos θ, y = 3 sin θ. Find the gradient of the curve at the point where θ = π/4.

  1. A
    −3/2
  2. B
    3/2
  3. C
    −2/3
  4. D
    2/3
Show answer

Answer: A. dx/dθ = −2 sin θ and dy/dθ = 3 cos θ, so dy/dx = −3 cos θ/(2 sin θ) = −(3/2) cot θ. At θ = π/4, cot θ = 1, so the gradient is −3/2.

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How many Differentiation MCQs are there?

32, each with the answer and a short explanation, across 1 topics: Differentiation. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 chapter 4 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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