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Differentiation: WJEC A‑Level Maths questions

Every WJEC A-Level Maths exam question on P.7 Differentiation, 2018–2025, with its mark scheme and examiner’s report.

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314 questions

  1. Q1

    Summer 2025 · Unit 3

    11 marksP.7 Differentiation
    The whole paper ↗
    Summer 2025 Unit 3, question 1: 1. (a) Differentiate each of the following functions with respect to x.   3 e x 2   x (i) [2] 2 x e sin3 x (ii) [3] cos2 x (iii) [3] 2 x  1 (b) A function

  2. Q5

    Summer 2025 · Unit 3

    8 marksP.7 Differentiation
    The whole paper ↗
    Summer 2025 Unit 3, question 5: 2 f x ( )  x  6 x  12 x  . 5 5. A curve is defined by (a) Find the coordinates of the stationary point and determine its nature. [6] Determine the range of

  3. Q13

    Summer 2025 · Unit 3

    11 marksP.7 DifferentiationP.3 Coordinate geometry
    The whole paper ↗
    Summer 2025 Unit 3, question 13: 13. A curve is defined parametrically by 2 3 1 y 3 t 2 x  t  ,  . Show that the gradient of the curve at the point with parameter p is 1 (a) p . [4] t  . 1

  4. Q6

    Summer 2025 · AS Unit 1

    16 marksP.2 Algebra and functionsP.7 Differentiation
    The whole paper ↗
    Summer 2025 AS Unit 1, question 6: 2 f x ( ) = x  3 x  6 x  . 8 6. The function f is defined by f x ( ) = 0 (a) Use an algebraic method to solve the equation . Show all your working. [6] y  f

  5. Q8

    Summer 2025 · AS Unit 1

    10 marksP.3 Coordinate geometryP.7 Differentiation
    The whole paper ↗
    Summer 2025 AS Unit 1, question 8: 2 2 4 4 x  y  x  y  . 8. A circle C has centre A and equation (a) Find the coordinates of A and the radius of C. [3] y  2 x  , intersects the circle C at

  6. Q9

    Summer 2025 · AS Unit 1

    5 marksP.7 Differentiation
    The whole paper ↗
    Summer 2025 AS Unit 1, question 9:  , find d y 2 2 y  x x 9. Given that x from first principles. [5] d

  7. Q11

    Summer 2025 · AS Unit 1

    9 marksP.7 DifferentiationP.2 Algebra and functions
    The whole paper ↗
    Summer 2025 AS Unit 1, question 11: f x ( ) x 3 9 x 2 28 x 4     . 11. The function f is defined by f x is an increasing function. ( ) (a) Show that [4] y = f x . ( ) (b) Two straight lines, b

  8. Q6

    Summer 2024 · Unit 3

    13 marksP.7 DifferentiationP.8 Integration
    The whole paper ↗
    Summer 2024 Unit 3, question 6: Differentiate cosx from first principles. 6. (a) [5] Differentiate 3e sin4 x x with respect to x. (b) [3]  2sin2 d x x x (c) Find . [5]

  9. Q11

    Summer 2024 · Unit 3

    10 marksP.7 DifferentiationP.3 Coordinate geometry
    The whole paper ↗
    Summer 2024 Unit 3, question 11: 11. A curve is defined parametrically by x  2 sin2 y  1 cos2   , . Show that the gradient of the curve at the point with parameter  is tan  . (a) [6]

  10. Q13

    Summer 2024 · Unit 3

    3 marksP.7 Differentiation
    The whole paper ↗
    Summer 2024 Unit 3, question 13:  for the interval    y  f x x ,x 13. The diagram below shows a sketch of the graph of . 1 5 y y = (x) x3 x x4 x5 x1 x2 O Find the interval on which  f x

  11. Q14

    Summer 2024 · Unit 3

    7 marksP.8 IntegrationP.7 Differentiation
    The whole paper ↗
    Summer 2024 Unit 3, question 14: d y 1 lnx ln x   y  14. (a) Given that  , show that . [2] x d x x 2 (b) Hence, solve the differential equation 2 d x x t ,  d t ln x 1 t  3 x  . given th

  12. Q1

    Summer 2024 · AS Unit 1

    4 marksP.7 Differentiation
    The whole paper ↗
    Summer 2024 AS Unit 1, question 1: , find the value of d y 27 y  12 x   4 x  9 1. Given that x when . [4] d x

  13. Q10

    Summer 2024 · AS Unit 1

    6 marksP.7 DifferentiationP.2 Algebra and functions
    The whole paper ↗
    Summer 2024 AS Unit 1, question 10: 10. Water is being emptied out of a sink. The depth of water, y cm, at time t seconds, may be modelled by y t 2 14 t 49    0 X t X 7. (a) Find the value of t

  14. Q12

    Summer 2024 · AS Unit 1

    10 marksP.7 Differentiation
    The whole paper ↗
    Summer 2024 AS Unit 1, question 12: y x 3 12 x 20    12. A curve C has equation . (a) Find the coordinates of the stationary points of C and determine their nature. [7] (b) Determine the range

  15. Q2

    Summer 2023 · Unit 3

    13 marksP.7 Differentiation
    The whole paper ↗
    Summer 2023 Unit 3, question 2: 2 a) Differentiate each of the following with respect to x. i)   5 sin x  x 2 [2] 3 cos x x ii) [2] 3 x e iii) [3] sin2 x b) Find the equation of the tangent

  16. Q11

    Summer 2023 · Unit 3

    7 marksP.7 Differentiation
    The whole paper ↗
    Summer 2023 Unit 3, question 11: ( ) = 5x3 + 2x2 − 3x . A curve C has equation f x 1 1 a) Find the x-coordinate of the point of inflection. State, with a reason, whether the point of inflection

  17. Q9

    Summer 2023 · AS Unit 1

    11 marksP.7 Differentiation
    The whole paper ↗
    Summer 2023 AS Unit 1, question 9: , find dy y x 2 3 x a) Given that   dx from first principles. [5] 0 9 3 6 The function f is defined by  f x  4 x  b) 2 for x > 0. x ( ). Find ′f i) x [2]

  18. Q14

    Summer 2023 · AS Unit 1

    6 marksP.6 Exponentials and logarithmsP.7 Differentiation
    The whole paper ↗
    Summer 2023 AS Unit 1, question 14: 4 a) Two variables, x and y, are such that the rate of change of y with respect to x is proportional to y. State a model which may be appropriate for y in terms

  19. Q2

    Summer 2022 · Unit 3

    5 marksP.7 Differentiation
    The whole paper ↗
    Summer 2022 Unit 3, question 2: 2 Differentiate the following functions with respect to x. x3ln(5x) a) [3] (x + cos3x)4 b) [2]

  20. Q13

    Summer 2022 · Unit 3

    7 marksP.7 Differentiation
    The whole paper ↗
    Summer 2022 Unit 3, question 13: A function is defined by f (x) = 2x3 + 3x – 5. 1 3 a) Prove that the graph of f (x) does not have a stationary point. [2] b) Show that the graph of f (x) does h
All 34 questions on Differentiation, as a list
  1. Summer 2025, Unit 3, Q1 (11 marks): 1. (a) Differentiate each of the following functions with respect to x.   3 e x 2   x (i) [2] 2 x e sin3 x (ii) [3] cos2 x (iii) [3] 2 x  1 (b) A function
  2. Summer 2025, Unit 3, Q5 (8 marks): 2 f x ( )  x  6 x  12 x  . 5 5. A curve is defined by (a) Find the coordinates of the stationary point and determine its nature. [6] Determine the range of
  3. Summer 2025, Unit 3, Q13 (11 marks): 13. A curve is defined parametrically by 2 3 1 y 3 t 2 x  t  ,  . Show that the gradient of the curve at the point with parameter p is 1 (a) p . [4] t  . 1
  4. Summer 2025, AS Unit 1, Q6 (16 marks): 2 f x ( ) = x  3 x  6 x  . 8 6. The function f is defined by f x ( ) = 0 (a) Use an algebraic method to solve the equation . Show all your working. [6] y  f
  5. Summer 2025, AS Unit 1, Q8 (10 marks): 2 2 4 4 x  y  x  y  . 8. A circle C has centre A and equation (a) Find the coordinates of A and the radius of C. [3] y  2 x  , intersects the circle C at
  6. Summer 2025, AS Unit 1, Q9 (5 marks):  , find d y 2 2 y  x x 9. Given that x from first principles. [5] d
  7. Summer 2025, AS Unit 1, Q11 (9 marks): f x ( ) x 3 9 x 2 28 x 4     . 11. The function f is defined by f x is an increasing function. ( ) (a) Show that [4] y = f x . ( ) (b) Two straight lines, b
  8. Summer 2024, Unit 3, Q6 (13 marks): Differentiate cosx from first principles. 6. (a) [5] Differentiate 3e sin4 x x with respect to x. (b) [3]  2sin2 d x x x (c) Find . [5]
  9. Summer 2024, Unit 3, Q11 (10 marks): 11. A curve is defined parametrically by x  2 sin2 y  1 cos2   , . Show that the gradient of the curve at the point with parameter  is tan  . (a) [6]
  10. Summer 2024, Unit 3, Q13 (3 marks):  for the interval    y  f x x ,x 13. The diagram below shows a sketch of the graph of . 1 5 y y = (x) x3 x x4 x5 x1 x2 O Find the interval on which  f x
  11. Summer 2024, Unit 3, Q14 (7 marks): d y 1 lnx ln x   y  14. (a) Given that  , show that . [2] x d x x 2 (b) Hence, solve the differential equation 2 d x x t ,  d t ln x 1 t  3 x  . given th
  12. Summer 2024, AS Unit 1, Q1 (4 marks): , find the value of d y 27 y  12 x   4 x  9 1. Given that x when . [4] d x
  13. Summer 2024, AS Unit 1, Q10 (6 marks): 10. Water is being emptied out of a sink. The depth of water, y cm, at time t seconds, may be modelled by y t 2 14 t 49    0 X t X 7. (a) Find the value of t
  14. Summer 2024, AS Unit 1, Q12 (10 marks): y x 3 12 x 20    12. A curve C has equation . (a) Find the coordinates of the stationary points of C and determine their nature. [7] (b) Determine the range
  15. Summer 2023, Unit 3, Q2 (13 marks): 2 a) Differentiate each of the following with respect to x. i)   5 sin x  x 2 [2] 3 cos x x ii) [2] 3 x e iii) [3] sin2 x b) Find the equation of the tangent
  16. Summer 2023, Unit 3, Q11 (7 marks): ( ) = 5x3 + 2x2 − 3x . A curve C has equation f x 1 1 a) Find the x-coordinate of the point of inflection. State, with a reason, whether the point of inflection
  17. Summer 2023, AS Unit 1, Q9 (11 marks): , find dy y x 2 3 x a) Given that   dx from first principles. [5] 0 9 3 6 The function f is defined by  f x  4 x  b) 2 for x > 0. x ( ). Find ′f i) x [2]
  18. Summer 2023, AS Unit 1, Q14 (6 marks): 4 a) Two variables, x and y, are such that the rate of change of y with respect to x is proportional to y. State a model which may be appropriate for y in terms
  19. Summer 2022, Unit 3, Q2 (5 marks): 2 Differentiate the following functions with respect to x. x3ln(5x) a) [3] (x + cos3x)4 b) [2]
  20. Summer 2022, Unit 3, Q13 (7 marks): A function is defined by f (x) = 2x3 + 3x – 5. 1 3 a) Prove that the graph of f (x) does not have a stationary point. [2] b) Show that the graph of f (x) does h
  21. Summer 2022, Unit 3, Q15 (9 marks): 5 A rectangle is inscribed in a semicircle with centre O and radius 4. The point P(x, y) is the vertex of the rectangle in the first quadrant as shown in the di
  22. Summer 2022, Unit 3, Q16 (8 marks): 6 The parametric equations of the curve C are x = 3 – 4t + t2, y = (4 – t)2. a) Find the coordinates of the points where C meets the y-axis. [3] b) Show that th
  23. Summer 2022, AS Unit 1, Q11 (15 marks): 1 The diagram below shows a sketch of the curve y = f (x), where f (x) = 10x + 3x2 − x3. The curve intersects the x-axis at the origin O and at the points A(–2,
  24. Summer 2022, AS Unit 1, Q14 (12 marks): ( ) = 3x3 − 5x2 + x − 6 1 4 A curve C has equation . f x a) Find the coordinates of the stationary points of C and determine their nature. [8] b) Without solvin
  25. Summer 2019, Unit 3, Q6 (11 marks): A curve C has parametric equations x = sinθ, y = cos2θ. 0 6 π The equation of the tangent to the curve C at the point P where θ= is a) 4 y = mx + c. Find the ex
  26. Summer 2019, Unit 3, Q10 (11 marks): a) Differentiate each of the following functions with respect to x. 1 0 i) x5 lnx e3x ii) x3 −1 1 ( tan x + 7x ) iii) 2 [8] b) A function is defined implicitly
  27. Summer 2019, AS Unit 1, Q8 (8 marks): dy Given that y = 2x2 – 5x, find from first principles. 0 8 a) [5] dx dy 1 y = 16 4 + 48 Given that , find the value of when x = 16. b) 5 x [3] dx x
  28. Summer 2019, AS Unit 1, Q13 (11 marks): y = 1 9 x3 − kx + 5 1 3 A curve C has equation . A point Q lies on C and is such that the tangent to C at Q has gradient –9. The x-coordinate of Q is 3. Show th
  29. Summer 2019, AS Unit 1, Q15 (4 marks): Prove that f(x) = x3 – 6x2 + 13x – 7 is an increasing function. 1 5 [4]
  30. Summer 2018, Unit 3, Q10 (14 marks): 0 The equation of a curve C is given by the parametric equations x = cos2θ , y = cosθ. a) Find the Cartesian equation of C. [2] π Show that the line x – y + 1 =
  31. Summer 2018, Unit 3, Q16 (11 marks): 6 a) Differentiate the following functions with respect to x, simplifying your answer wherever possible. e3tanx, i) sin2x ii) . [5] x2 b) A function is defined
  32. Summer 2018, AS Unit 1, Q4 (6 marks): dy y = 5 x + 6 x 0 4 a) Given that , find when x = 8. [3] 3 dx (5x 2 +12x−5 + 7)dx 3 ∫ b) Find . [3]
  33. Summer 2018, AS Unit 1, Q13 (11 marks): 3 A curve C has equation y = x3 – 3x2. a) Find the stationary points of C and determine their nature. [7] b) Draw a sketch of C, clearly indicating the stationa
  34. Summer 2018, AS Unit 1, Q16 (5 marks): The curve C has equation y = 7 + 13x – 2x2. The point P lies on C and is such that the 1 6 tangent to C at P has equation y = x + c, where c is a constant. Find

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