plumiDifferentiation: OCR MEI A‑Level Maths questions
Every OCR MEI A-Level Maths exam question on P.7 Differentiation, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 1, question 10: x2 e 10 The diagram shows part of the graph of the function y + which is defined for x 2- . 1 = x 1 y 4 3 2 1 0 x –1 1 d . y (a) Find an expression for x [3] d](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun25-10-qp.webp?v=3)
![June 2025 Paper 1, question 11: The curve in the graph below is defined implicitly by ( 2 x y )( y 1 ) 6 . + - = y x d = y 2 1 ( y ) - (a) Show that . [4] d x 2 y 2 x 1 + - (b) Find the equati](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun25-11-qp.webp?v=3)
![June 2025 Paper 2, question 14: 3 14 The equation of a curve is y + x . = 2 x (a) Determine the coordinates of the point where the curve cuts the x-axis. [2] d . y (b) (i) Find x [2] d 16 3 (i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun25-14-qp.webp?v=3)
![June 2025 Paper 3, question 9: 3 x 9 The function ( ) f x is defined on all real numbers by ( ) f x x e . = + 3 3 x (a) Differentiate x e . [2] + (b) Write down the coordinates of the point w](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun25-09-qp.webp?v=3)
![June 2025 AS Paper 1, question 11: In this question you must show detailed reasoning. 2 f x x x x You are given that ( ) ( 2 )( 7 6 ) . = + - + (a) Solve the equation ( ) f x [3] 0 . = (b) Show t](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63001-jun25-11-qp.webp?v=3)
![June 2025 AS Paper 2, question 6: The equation of a curve is 4 3 y x 8 x 20 . = - + d , is y (a) Show that the gradient function, x d d y 2 4 x ( x 6 ) . [2] = - d x (b) (i) On the axes in the P](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63002-jun25-06-qp.webp?v=3)
![June 2024 Paper 1, question 6: 6 Given that ( )x f 2 x + , show from first principles that 3 f ( )x 4 x . [4] l = =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun24-06-qp.webp?v=3)
![June 2024 Paper 1, question 13: 13 The curve with equation y px + , where p and q are constants, has a stationary point q = + 2 x at (2, 7). (a) Determine the values of p and q. [5] 2 d y (b)](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64001-jun24-13-qp.webp?v=3)
![June 2024 Paper 2, question 8: The equation of a curve is 3 2 y 2 x 3 mx 9 mx + . 4 = + - Determine the range of values of m for which the curve has no stationary values. [6]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun24-08-qp.webp?v=3)
![June 2024 Paper 2, question 13: Determine the coordinates of the turning points on the curve with equation 2 2 y xy x x = . 1 [9] + + -](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64002-jun24-13-qp.webp?v=3)
![June 2024 Paper 3, question 15: 15 (a) Show that, for the curve y ax bx + , the equation of the tangent at the point with c = + 2 x‑coordinate t is y ( 2 at b x ) at + . c [3] = + - 2 (b) Henc](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun24-15-qp.webp?v=3)
![June 2024 Paper 3, question 17: x 1 x2 t t2 is y 2 17 Show that, for the curve y , the equation of the normal at the point ( , ) t , = =- + + 2 t 2 as given in line 27. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h64003-jun24-17-qp.webp?v=3)
![June 2024 AS Paper 1, question 5: In this question you must show detailed reasoning. a k is c 1 m at the point 1 31 99 , (a) Show that the gradient of the curve y x x 2 x . [4] = - - 2 4 4 2 a k](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h63001-jun24-05-qp.webp?v=3)
All 69 questions on Differentiation, as a list
- June 2025, Paper 1, Q10 (10 marks): x2 e 10 The diagram shows part of the graph of the function y + which is defined for x 2- . 1 = x 1 y 4 3 2 1 0 x –1 1 d . y (a) Find an expression for x [3] d
- June 2025, Paper 1, Q11 (10 marks): The curve in the graph below is defined implicitly by ( 2 x y )( y 1 ) 6 . + - = y x d = y 2 1 ( y ) - (a) Show that . [4] d x 2 y 2 x 1 + - (b) Find the equati
- June 2025, Paper 2, Q14 (15 marks): 3 14 The equation of a curve is y + x . = 2 x (a) Determine the coordinates of the point where the curve cuts the x-axis. [2] d . y (b) (i) Find x [2] d 16 3 (i
- June 2025, Paper 3, Q6 (3 marks): The diagram shows the curve with equation y f ( ) x , where ( ) f x 13 x 34 ln ( ) x 1 5 . for = = - - x 0 . 2 y 5 4 3 2 1 0 x 0 1 2 3 4 5 (a) A student attempt
- June 2025, Paper 3, Q8 (3 marks): The diagram below shows the graph of y as a function of x. Point D is the fixed point on the graph where x 10 . = Point E is the point on the graph where x 10 +
- June 2025, Paper 3, Q9 (5 marks): 3 x 9 The function ( ) f x is defined on all real numbers by ( ) f x x e . = + 3 3 x (a) Differentiate x e . [2] + (b) Write down the coordinates of the point w
- June 2025, AS Paper 1, Q11 (9 marks): In this question you must show detailed reasoning. 2 f x x x x You are given that ( ) ( 2 )( 7 6 ) . = + - + (a) Solve the equation ( ) f x [3] 0 . = (b) Show t
- June 2025, AS Paper 2, Q6 (6 marks): The equation of a curve is 4 3 y x 8 x 20 . = - + d , is y (a) Show that the gradient function, x d d y 2 4 x ( x 6 ) . [2] = - d x (b) (i) On the axes in the P
- June 2025, AS Paper 2, Q12 (4 marks): 12 The equation of a curve is ( ) f x 3 x 2 x . = + (a) Determine an expression for the gradient of the chord joining the points ^ h and ^ h. Express your answe
- June 2024, Paper 1, Q6 (4 marks): 6 Given that ( )x f 2 x + , show from first principles that 3 f ( )x 4 x . [4] l = =
- June 2024, Paper 1, Q13 (8 marks): 13 The curve with equation y px + , where p and q are constants, has a stationary point q = + 2 x at (2, 7). (a) Determine the values of p and q. [5] 2 d y (b)
- June 2024, Paper 1, Q16 (7 marks): A block of mass m kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is n. A force of magnitude T N is applied at
- June 2024, Paper 2, Q8 (6 marks): The equation of a curve is 3 2 y 2 x 3 mx 9 mx + . 4 = + - Determine the range of values of m for which the curve has no stationary values. [6]
- June 2024, Paper 2, Q13 (9 marks): Determine the coordinates of the turning points on the curve with equation 2 2 y xy x x = . 1 [9] + + -
- June 2024, Paper 3, Q10 (3 marks): The diagram below shows the curve y f ( ) x . = y x 0 –4 –3 –2 –1 1 2 3 4 Sketch the graph of the gradient function, y f ( ) x , on the copy of the diagram in t
- June 2024, Paper 3, Q11 (8 marks): Fig. 11.1 shows the curve with equation y g ( ) x where ( ) g x x sin x cos x and the curve of the = = + gradient function y g ( ) x for 2 x 2 . = l - r r G G F
- June 2024, Paper 3, Q12 (9 marks): The diagram shows the curve with parametric equations x sin 2 + , 2 y 2 cos cos 2 for 0 1 2 . , r = i = i + i G i y x O (a) In this question you must show detai
- June 2024, Paper 3, Q15 (6 marks): 15 (a) Show that, for the curve y ax bx + , the equation of the tangent at the point with c = + 2 x‑coordinate t is y ( 2 at b x ) at + . c [3] = + - 2 (b) Henc
- June 2024, Paper 3, Q17 (3 marks): x 1 x2 t t2 is y 2 17 Show that, for the curve y , the equation of the normal at the point ( , ) t , = =- + + 2 t 2 as given in line 27. [3]
- June 2024, AS Paper 1, Q5 (6 marks): In this question you must show detailed reasoning. a k is c 1 m at the point 1 31 99 , (a) Show that the gradient of the curve y x x 2 x . [4] = - - 2 4 4 2 a k
- June 2024, AS Paper 1, Q12 (10 marks): The diagram shows the graph of ( ) f x k x ( p x )( q ) where k , p and q are constants. The graph = - - passes through the points (–1, 0), (0, –4) and (2, 0).
- June 2024, AS Paper 2, Q13 (5 marks): 2 13 Determine the range of values of x for which y 4 x 7 x 6 x + is a decreasing function. 8 [5] = + -
- June 2023, Paper 1, Q5 (5 marks): In this question you must show detailed reasoning. 16 2 (a) Find the coordinates of the two stationary points on the graph of y 15 x . [3] = - - 2 x (b) Show th
- June 2023, Paper 2, Q15 (7 marks): In this question you must show detailed reasoning. The equation of a curve is 3 ln y x y 8 . + = Find the equation of the normal to the curve at the point where
- June 2023, Paper 3, Q5 (8 marks): In this question you must show detailed reasoning. 3 2 This question is about the curve y x 5 x 6 x . = - + (a) Find the equation of the tangent, T, to the curv
- June 2023, Paper 3, Q7 (6 marks): A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is r cm and the angle at the centre is i radians.
- June 2023, AS Paper 2, Q11 (5 marks): In this question you must show detailed reasoning. 3 2 The equation of a curve is y 2 x 9 x 24 x - . 8 = + + Show that there are no stationary points on this cu
- June 2023, AS Paper 2, Q14 (7 marks): In this question you must show detailed reasoning. 8 The equation of a curve is y 16 x . = + x Determine the equation of the tangent to the curve at the point w
- June 2022, Paper 1, Q8 (10 marks): t2 8 A particle moves in the x-y plane so that its position at time t s is given by x t t8 , y for = - = 3 5 . 1 t 1 3 5 . . The units of distance are metres. T
- June 2022, Paper 1, Q10 (8 marks): A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins B and C. AB is a 30 cm rod and AC is a 15 cm rod. The angle BA
- June 2022, Paper 2, Q16 (15 marks): The equation of a curve is 4 3 2 y x x x x 6 8 21 12 6 . = + - + - (a) In this question you must show detailed reasoning. Determine • The coordinates of the sta
- June 2022, Paper 3, Q5 (7 marks): 2 5 A curve is defined implicitly by the equation x 2 3 xy y 2 0 . + + + = d y 4 x 3 y + (a) Show that d . [3] =- x y x 3 2 + (b) In this question you must show
- June 2022, Paper 3, Q9 (2 marks): Show that y = has the same gradient as x y sin x when x 0 , as stated in line 5. [2] = =
- June 2022, Paper 3, Q10 (5 marks): In this question you must show detailed reasoning. 4 x ( x ) - r Fig. C2.2 indicates that the curve y sin x has a stationary point near x = . 3 = - 2 r • Verify
- June 2022, AS Paper 1, Q12 (6 marks): x2 12 Below is a faulty argument that appears to show that the gradient of the curve y at the point = (3, 9) is 1. Consider the chord joining (3, 9) to the poin
- June 2022, AS Paper 2, Q9 (10 marks): 9 The equation of a curve is y 12 x 4 x . = - 2 (a) State the coordinates of the intersection of the curve with the y-axis. [1] (b) Find the value of y when x 9
- November 2021, Paper 1, Q6 (7 marks): (a) The diagram shows part of the graph of y cosec x , where x is in radians. = State the equations of the three vertical asymptotes that can be seen. [1] y x O
- November 2021, Paper 1, Q11 (11 marks): A balloon is being inflated. The balloon is modelled as a sphere with radius x cm at time t s. The 4 r volume V cm3 is given by V x3 . = 3 The rate of increase
- November 2021, Paper 2, Q14 (13 marks): The equation of a curve is 2 3 y x ( x 2 ) . = - d , giving your answer in factorised form. y (a) Find x [4] d (b) Determine the coordinates of the stationary p
- November 2021, Paper 3, Q4 (3 marks): x c m - x 4 The diagram shows points A and B on the curve y . = 4 The x-coordinate of A is 1 and the x-coordinate of B is 1.1. y B A x O (a) Find the gradient o
- November 2021, AS Paper 1, Q8 (12 marks): In this question you must show detailed reasoning. (a) Use differentiation to find the coordinates of the stationary point on the curve with equation 2 y 2 x 3
- November 2021, AS Paper 2, Q13 (9 marks): In this question you must show detailed reasoning. 7 3 The equation of a curve is y 3 x . = + - x 2 x Determine the coordinates of the points on the curve where
- November 2020, Paper 1, Q4 (5 marks): ^ h , giving your answer in factorised form. 4 2 4 Find the second derivative of x 5 [5] +
- November 2020, Paper 1, Q10 (9 marks): In this question you must show detailed reasoning. 1 1 1 Fig. 10 shows the curve given parametrically by the equations x , y , for t 2 0 . = = - t 2 3 t t y x O
- November 2020, Paper 1, Q12 (9 marks): 12 A function is defined by ( )x f x - . x = x h x (a) By considering ( f ) f ( ) + - 2 , show from first principles that f ( )x 3 x 1 . [4] l = - h (b) Sketch
- November 2020, Paper 2, Q3 (4 marks): You are given that y 4 x sin 8 x . = + d . y (a) Find x [2] d d = y (b) Find the smallest positive value of x for which x 0 , giving your answer in an exact for
- November 2020, Paper 2, Q15 (7 marks): Functions f (x) and g (x) are defined as follows. 3 f ( )x x for x 2 0 and ( )x g x - - for x 2 2. x 6 = = The function h (x) is defined as h (x) = fg (x). (a)
- November 2020, Paper 3, Q8 (16 marks): 8 (a) The curve y is shown in Fig. 8. ^ h = 2 2 x 1 + y x O Fig. 8 2 2 d y 20 x 4 - (i) Show that . [5] = 2 2 4 d x ( 1 x ) + (ii) In this question you must sho
- November 2020, Paper 3, Q11 (2 marks): Show that ex is an increasing function for all values of x, as stated in line 39. [2]
- November 2020, Paper 3, Q12 (8 marks): ln x 12 (a) Show that the only stationary point on the curve y occurs where x = e, as given in = x line 45. [3] (b) Show that the stationary point is a maximum.
- November 2020, AS Paper 1, Q5 (4 marks): 5 Fig. 5.1 shows part of the curve y x2 . P is the point (1, 1) and Q is the point on the curve with = x-coordinate 1 +h. y Q P x O Fig. 5.1 Table 5.2 shows, fo
- November 2020, AS Paper 1, Q7 (6 marks): In this question you must show detailed reasoning. 3 2 A curve has equation y 4 x 6 x 9 x + . 4 = - - (a) Sketch the gradient function for this curve, clearly i
- November 2020, AS Paper 1, Q12 (12 marks): In this question you must show detailed reasoning. 1 2 Fig. 12 shows part of the graph of y x . = + 2 x y x O Fig. 12 1 2 17 meets the x-axis at A and meets the
- November 2020, AS Paper 2, Q9 (8 marks): 9 The equation of a curve is y 24 x 8 x 16 . = - + 2 d . y (a) Find x [3] d (b) Find the coordinates of the turning point. [3] (c) Determine the nature of the t
- November 2020, AS Paper 2, Q11 (10 marks): - . 11 A car is travelling along a stretch of road at a steady speed of 11ms 1 The driver accelerates, and t seconds after starting to accelerate the speed of t
- June 2019, Paper 2, Q2 (4 marks): 12 2 Given that y ( x 5 ) , = + y d . (a) Find x [2] d ; 2 11 (b) Hence find 48 x x ( 5 ) d x . [2] +
- June 2019, Paper 2, Q11 (8 marks): 11 Fig. 11 shows the graph of y x 4 x x ln x . = - + y 8 6 4 2 x 0 2 4 6 –2 Fig. 11 (a) Show that the x-coordinate of the stationary point on the curve may be f
- June 2019, Paper 3, Q8 (10 marks): In this question you must show detailed reasoning. x3 A is the point (1, 0), B is the point (1, 1) and D is the point where the tangent to the curve y at = B cr
- June 2019, Paper 3, Q9 (6 marks): In this question you must show detailed reasoning. 2 The curve xy y 8 is shown in Fig. 9. + = y x O Fig. 9 [6] Find the coordinates of the points on the curve a
- June 2019, Paper 3, Q10 (4 marks): x e 10 Show that ( ) is an increasing function for all values of x. [4] f x = x 1 e +
- June 2019, AS Paper 1, Q8 (7 marks): In this question you must show detailed reasoning. 2 Show that the only stationary point on the graph of y x 4 x is a minimum point at (,1 - 3 ) . = - [7]
- June 2019, AS Paper 1, Q10 (7 marks): In this question you must show detailed reasoning. 3 2 3 (a) Sketch the gradient function for the curve y 24 x x x . [5] = - - 3 2 3 (b) Determine the set of va
- June 2019, AS Paper 2, Q10 (10 marks): In this question you must show detailed reasoning. 2 x 2 The equation of a curve is y + . A tangent and a normal to the curve are drawn at the 1 = + x 4 point w
- June 2018, Paper 1, Q13 (12 marks): 8 3 13 The function f (x) is defined by ( ) f x 27 x . Jenny uses her scientific calculator to create a table of = - values for f (x) and f x ( ) . l x f x ( )
- June 2018, Paper 2, Q15 (9 marks): You must show detailed reasoning in this question. The equation of a curve is 3 - y xy 4 x 4 . + = Find the gradient of the curve at each of the points where y
- June 2018, Paper 3, Q4 (10 marks): In this question you must show detailed reasoning. 1 A curve has equation y x 5 - . The curve is shown in Fig. 4. = - + x 2 y x O Fig. 4 (i) Determine the coord
- June 2018, Paper 3, Q8 (8 marks): t t 8 A curve has parametric equations x , y t , where 1 . ! - = = 3 3 1 t 1 t + + (i) In this question you must show detailed reasoning. Determine the gradient
- June 2018, AS Paper 1, Q9 (9 marks): ^ h maps onto the curve C1 following a stretch scale factor 2 1 2 1 in the x-direction. 9 The curve y x = - 2 (i) Show that the equation of C1 can be written as
- June 2018, AS Paper 2, Q10 (9 marks): 10 (i) A curve has equation y 16 x . Find = + 2 x d , y (A) [2] x d 2 d y (B) . [2] 2 d x (ii) Hence • find the coordinates of the stationary point, • determine
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