plumiDifferentiation: OCR A‑Level Maths questions
Every OCR 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 2, question 6: 3 2 6 A curve has equation y 3 x 4 x 6 x 12 x . = - - + d . y (a) (i) Find x [2] d (ii) Use your answer to part (a)(i) to find the x-coordinates of the stationa](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun25-06-qp.webp?v=3)
![June 2025 AS Paper 2, question 4: 2 4 The cubic polynomial x 2 kx 4 x + , where k is a constant, is denoted by ( )x k f . - + It is given that f ( ) 2 16 . l = (a) Show that k = 3. [4] For the r](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23002-jun25-04-qp.webp?v=3)
![June 2024 Paper 1, question 5: 5 The line x 13 y 108 is the normal to the curve y ax b x at the point (4, 8). + = = + Determine the values of the constants a and b. [8]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun24-05-qp.webp?v=3)
![June 2024 Paper 1, question 11: A curve has equation y 5 ln ( 1 cos 2 x ) , where x is in radians. = - (a) State the values of x for which 5 ln ( 1 cos x 2 ) is not defined. [2] - (b) P is the](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun24-11-qp.webp?v=3)
![June 2024 Paper 2, question 1: Differentiate the following with respect to x. 2 4 (a) 3 x [2] - 2 x (b) 4 x - 9 [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun24-01-qp.webp?v=3)
![June 2024 AS Paper 1, question 5: 2 5 A curve has equation y 4 x . = + x (a) Determine the coordinates of the stationary point on the curve. [5] 2 d y (b) Use d to determine whether this turning](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun24-05-qp.webp?v=3)
![June 2023 Paper 1, question 3: 3 (a) Given that ( ) f x x 2 x , use differentiation from first principles to show that f ( )x 2 x 2 . l = + = + [4] d = y (b) The gradient of a curve is given](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-03-qp.webp?v=3)
![June 2023 Paper 1, question 6: + . ex 2 3 x 6 A curve has equation y = (a) Determine the x-coordinates of any stationary points on the curve. [4] (b) Show that the curve is convex for all val](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-06-qp.webp?v=3)
All 72 questions on Differentiation, as a list
- June 2025, Paper 1, Q2 (4 marks): y x O The diagram shows the curve y ax x ( b ) , where a and b are constants and b 0 . = - 2 (a) Given that the curve has a stationary point at x = , state the
- June 2025, Paper 1, Q7 (9 marks): In this question you must show detailed reasoning. 3 2 2 A curve has parametric equations x t t , y t t2 for all real values of t. = + = + The curve passes thro
- June 2025, Paper 1, Q10 (12 marks): ex e x 2 1 10 The graph of y can be transformed to the graph of y - by a stretch parallel to the = = x-axis followed by a translation. (a) (i) State the scale f
- June 2025, Paper 2, Q6 (7 marks): 3 2 6 A curve has equation y 3 x 4 x 6 x 12 x . = - - + d . y (a) (i) Find x [2] d (ii) Use your answer to part (a)(i) to find the x-coordinates of the stationa
- June 2025, AS Paper 1, Q8 (8 marks): 2 8 A curve with equation y x 3 x 9 x 12 has stationary points A and B. = + - - Determine the coordinates of the point where the line through A and B meets the
- June 2025, AS Paper 2, Q4 (10 marks): 2 4 The cubic polynomial x 2 kx 4 x + , where k is a constant, is denoted by ( )x k f . - + It is given that f ( ) 2 16 . l = (a) Show that k = 3. [4] For the r
- June 2024, Paper 1, Q5 (8 marks): 5 The line x 13 y 108 is the normal to the curve y ax b x at the point (4, 8). + = = + Determine the values of the constants a and b. [8]
- June 2024, Paper 1, Q11 (12 marks): A curve has equation y 5 ln ( 1 cos 2 x ) , where x is in radians. = - (a) State the values of x for which 5 ln ( 1 cos x 2 ) is not defined. [2] - (b) P is the
- June 2024, Paper 2, Q1 (4 marks): Differentiate the following with respect to x. 2 4 (a) 3 x [2] - 2 x (b) 4 x - 9 [2]
- June 2024, Paper 2, Q4 (10 marks): - . 1 3 x 4 The diagram shows part of the graph of y xe = y 0.5 x 0 0.5 1 1.5 2 (a) Use the sign change method to determine, correct to 2 decimal places, the ro
- June 2024, Paper 3, Q5 (13 marks): y x O M ^ h 3 2 y x x x The diagram shows the curve with equation 2 ln . The curve has a point of inflection = - at the point M. (a) (i) Show that the x-coordin
- June 2024, Paper 3, Q7 (7 marks): y Q P x O 2 The diagram shows the curve x 5 2 xy 2 y k 0 , where k is a positive integer. - + - = At the points P and Q on the curve, the tangents to the curve
- June 2024, AS Paper 1, Q5 (9 marks): 2 5 A curve has equation y 4 x . = + x (a) Determine the coordinates of the stationary point on the curve. [5] 2 d y (b) Use d to determine whether this turning
- June 2024, AS Paper 2, Q5 (7 marks): e x 3 5 At the point P on the curve y 21 x - the gradient of the tangent is 3. 8 = - (a) Determine the x-coordinate of P. Give your answer in the form ln a, whe
- June 2024, AS Paper 2, Q6 (7 marks): y x O 2 P 5 4 3 The diagram shows the curve with equation y x ax bx , where a and b are integers. The = + + curve has a minimum at the point P where x 2 . = The
- June 2023, Paper 1, Q3 (7 marks): 3 (a) Given that ( ) f x x 2 x , use differentiation from first principles to show that f ( )x 2 x 2 . l = + = + [4] d = y (b) The gradient of a curve is given
- June 2023, Paper 1, Q6 (9 marks): + . ex 2 3 x 6 A curve has equation y = (a) Determine the x-coordinates of any stationary points on the curve. [4] (b) Show that the curve is convex for all val
- June 2023, Paper 1, Q9 (6 marks): Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a se
- June 2023, Paper 2, Q4 (9 marks): x2 4 The diagram shows part of the graph of y . The normal to the curve at the point A(1, 1) meets = the curve again at B. Angle AOB is denoted by a. y B 2 A 1
- June 2023, Paper 2, Q5 (12 marks): In this question you must show detailed reasoning. The function f is defined by f( ) x cos x 3 sin x with domain 0 x 2 r. = + G G (a) Solve the following equati
- June 2023, Paper 3, Q3 (8 marks): 3 The cubic polynomial ( ) f x is defined by ( )x f x px + , where p and q are constants. q = + (a) (i) Given that f 2 ( ) 13 , find the value of p. [2] l = (ii
- June 2023, AS Paper 1, Q3 (4 marks): The diagram shows the graph of y f( ) x , where f( )x is a quadratic function of x. = A copy of the diagram is given in the Printed Answer Booklet. y x (a) On t
- June 2023, AS Paper 1, Q4 (4 marks): x 4 A curve has equation y e . = (a) Determine the value of x when y 10 . [2] = (b) Determine the gradient of the tangent to the curve at the point where x 2 .
- June 2023, AS Paper 1, Q6 (11 marks): 3 6 (a) Determine the two real roots of the equation x x [3] 8 7 1 0 . + - = 7 49 4 (b) Determine the coordinates of the stationary points on the curve y x x x
- June 2023, AS Paper 2, Q7 (7 marks): y l2 C l1 1a k , 3 2 x O 2 The diagram shows the curve C with equation y 4 x 10 x + and two straight lines, l1 and l2. 7 = - 1a k. The line l2 is the normal to
- June 2022, Paper 1, Q7 (8 marks): 2 7 A curve has equation x 2 6 xy 3 y 2 . + - = Show that there are no points on this curve where the tangent is parallel to y = . x [8]
- June 2022, Paper 1, Q8 (9 marks): (a) Substance A is decaying exponentially such that its mass is m grams at time t minutes. Find the missing values of m and t in the following table. t 0 10 50
- June 2022, Paper 1, Q12 (12 marks): c m. 1 1 2 12 A curve has parametric equations x t y , 2 t . The point P is ,p p = = 2 (a) Show that the equation of the tangent at P can be written as y 2 p x
- June 2022, Paper 2, Q5 (4 marks): In this question you must show detailed reasoning. 3 2 A curve has equation y x 3 x 4 x . = - + (a) Show that the curve has no stationary points. [2] (b) Show t
- June 2022, Paper 2, Q8 (7 marks): 50cm 30° hcm The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle 30° and height 50 cm. Initially the tank is full, and t
- June 2022, Paper 3, Q5 (14 marks): In this question you must show detailed reasoning. y x O P x 2 3 - The diagram shows the curve with equation y . The tangent to the curve at the point P = 2 4 x
- June 2022, AS Paper 2, Q5 (7 marks): y x O 1 ( , 1 ) - 2 The diagram shows a curve C for which y is inversely proportional to x. The curve passes through a k. 1 the point ,1 - 2 (a) (i) Determine t
- June 2022, AS Paper 2, Q8 (11 marks): + 8 (a) The quadratic polynomial ax bx , where a and b are constants, is denoted by ( ) f x . Use differentiation from first principles to determine, in terms o
- November 2021, Paper 1, Q5 (7 marks): (a) The graph of the function y f ( ) x passes through the point P with coordinates (2, 6), and = is a one-one function. State the coordinates of the point corr
- November 2021, Paper 1, Q7 (9 marks): 7 The curve y ( x 2 )ln x has one stationary point which is close to x = . 1 = - 2 2 (a) Show that the x-coordinate of this stationary point satisfies the equat
- November 2021, Paper 1, Q9 (10 marks): A particle moves in the x-y plane so that at time t seconds, where t 0 , its coordinates are given by H - . 2 t t 3 t x e 4 e 3 , y 2 e = - + = (a) Explain why
- November 2021, Paper 1, Q11 (12 marks): 4 x y 4 2 2 2 2 11 (a) Use the substitution u x + to show that 3 d x ( x 6 ) x 3 c . [5] = = - + + 3 2 x 3 + (b) In this question you must show detailed reasoni
- November 2021, Paper 2, Q1 (4 marks): Differentiate the following with respect to x. x 4 (a) e - [2] x2 (b) + [2] x 1
- November 2021, Paper 2, Q7 (4 marks): Differentiate cos x with respect to x, from first principles. [4]
- November 2021, Paper 3, Q8 (11 marks): y L R P M O x 2 x The diagram shows the curve M with equation y xe - . = (a) Show that M has a point of inflection at the point P where x = . 1 [5] The line L p
- November 2021, AS Paper 1, Q2 (4 marks): The number of people, n, living in a small town is changing over time. In an attempt to predict the future growth of the town, a researcher uses the following m
- November 2021, AS Paper 1, Q5 (8 marks): The fuel consumption of a car, C miles per gallon, varies with the speed, v miles per hour. Jamal models the fuel consumption of his car by the formula 3 12 2 C
- November 2021, AS Paper 2, Q4 (5 marks): - is denoted by f (x). 4 The quadratic polynomial x 2 3 Use differentiation from first principles to determine the value of f ′(2). [5]
- November 2021, AS Paper 2, Q6 (6 marks): y d 2 6 A curve C has an equation which satisfies x 3 x + , for all values of x. 2 = 2 d (a) It is given that C has a single stationary point. Determine the nat
- November 2021, AS Paper 2, Q8 (11 marks): y l ,16 ^ h x O 6 The diagram shows the curve y 1 x and the line l, which is the normal to the curve at the = - + x point (1, 6). (a) Determine the equation of
- November 2020, Paper 1, Q3 (10 marks): A cylindrical metal tin of radius r cm is closed at both ends. It has a volume of 16000 cm3 3 . r - . (a) Show that its total surface area, Acm2, is given by A
- November 2020, Paper 1, Q8 (7 marks): e x 2 8 (a) Differentiate ( 2 3 x ) with respect to x. [3] + 2 e x 2 (b) Hence show that ( 2 3 x ) is increasing for all values of x. [4] +
- November 2020, Paper 2, Q1 (9 marks): (a) Differentiate the following with respect to x. 3 7 (i) ( 2 x ) [2] + 3 (ii) x ln x [3] y (b) Find cos x x 5 d . [2] d = y (c) Find the equation of the curve
- November 2020, Paper 3, Q4 (11 marks): 4 A curve has equation y 2 ln ( k 3 x ) x 3 x , where k is a positive constant. = - + - (a) Given that the curve has a point of inflection where x = 1, show tha
- November 2020, Paper 3, Q5 (12 marks): y P R C x O 1 6 The diagram shows the curve C with parametric equations 3 - , where t 3 2 t x , y t e 2 0 . = = t The maximum point on C is denoted by P. (a) De
- November 2020, Paper 3, Q6 (11 marks): In this question you must show detailed reasoning. y P x O Q 2 2 The diagram shows the curve with equation 4 xy 2 ( x 4 y ) 9 x . = + - y x y d = 4 4 - . 9 - (a
- November 2020, AS Paper 1, Q1 (6 marks): d c 5 m. 3 1 (a) Find x x 3 x [3] - + d 2 x c 2 d m y 2 (b) Find 6 x x . [3] - 3 x
- November 2020, AS Paper 2, Q3 (7 marks): In this question you must show detailed reasoning. Find the equation of the normal to the curve y 4 x 3 x + at the point on the curve where 1 = - x 4 . Give you
- November 2020, AS Paper 2, Q7 (13 marks): E xcm B D F A C 2xcm The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism ABCDEF. The base ACFD of the
- June 2019, Paper 1, Q5 (8 marks): 2 2 5 A curve has equation x 3 x y y 1 0 . - + + = 2 y xy x d 6 3 - (a) Show that x . [4] = d 2 2 y 3 x - (b) Find the equation of the normal to the curve at th
- June 2019, Paper 1, Q6 (6 marks): 2 6 Let ( )x f 2 x 3 x . Use differentiation from first principles to show that f ( )x 6 x 3 . [6] = + l = +
- June 2019, Paper 1, Q12 (12 marks): a x 3 2 12 A curve has equation y , where a is a constant greater than 1. = y d 3 2 x (a) Show that 6 xa ln a . [3] = x d 3 passes through the point ( 2 1 , 0).
- June 2019, Paper 2, Q1 (11 marks): (a) Differentiate the following. 2 x (i) + [3] x 2 1 2 - (ii) tan x ( 3 x ) [2] 1 (b) Use the substitution u x - to integrate 1 - . [4] = x 1 x 2 - (c) Integrat
- June 2019, Paper 2, Q3 (9 marks): (a) A circle is defined by the parametric equations x 3 2 cos , y 4 2 sin . = + =- + i i (i) Find a cartesian equation of the circle. [2] (ii) Write down the ce
- June 2019, Paper 2, Q4 (5 marks): A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two p
- June 2019, Paper 2, Q5 (9 marks): h l r For a cone with base radius r, height h and slant height l, the following formulae are given. Curved surface area, S rl = r 1 2 Volume, V r h = r 3 A cont
- June 2019, Paper 3, Q4 (14 marks): y P x O 1 r 2 1 The diagram shows the part of the curve y 3 x sin 2 x for which 0 x = G G . r 2 The maximum point on the curve is denoted by P. (a) Show that th
- June 2019, AS Paper 1, Q1 (8 marks): 1 It is given that ( )x f 3 x . = - 3 x Find (a) f l ( )x , [3] (b) f ll ( )x , [2] y (c) f ( )x d x . [3]
- June 2019, AS Paper 1, Q4 (6 marks): 2 4 (a) Find the coordinates of the stationary points on the curve y x 6 x 9 x . [4] = - + 3 2 (b) The equation x 6 x 9 x k 0 has exactly one real root. - + + =
- June 2019, AS Paper 1, Q7 (11 marks): ekx 7 (a) Write down an expression for the gradient of the curve y . [1] = 1 e x (b) The line L is a tangent to the curve y at the point where x 2 . = = 2 Show
- June 2018, Paper 1, Q10 (10 marks): 2 10 A curve has parametric equations x t and y t , for t 0 . ! = + = - t t d in terms of t, giving your answer in its simplest form. y (i) Find x [4] d (ii) Ex
- June 2018, Paper 1, Q11 (11 marks): In a science experiment a substance is decaying exponentially. Its mass, M grams, at time t minutes is given 0 05 . t by M e 300 - . = (i) Find the time taken f
- June 2018, Paper 1, Q12 (10 marks): In this question you must show detailed reasoning. y x 1 r ( , ) 0 4 4 cos 2 x 1 r The diagram shows the curve y , for x 0 , and the normal to the curve at the
- June 2018, Paper 2, Q6 (13 marks): In this question you must show detailed reasoning. ln x y A curve has equation . = x (i) Find the x-coordinate of the point where the curve crosses the x axis.
- June 2018, AS Paper 1, Q4 (9 marks): 4 (i) It is given that y x 3 x . = + d . y (a) Find x [2] d (b) Find the values of x for which y is increasing. [2] ^ h y (ii) Find 3 4 x d x . [5] -
- June 2018, AS Paper 1, Q6 (7 marks): Sketch the following curves. 2 (i) y [2] = x 3 2 (ii) y x 6 x 9 x [5] = - +
- June 2018, AS Paper 2, Q6 (9 marks): A pan of water is heated until it reaches 100 °C. Once the water reaches 100 °C, the heat is switched off and the temperature T °C of the water decreases. The t
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