plumiDifferentiation: Edexcel A‑Level Maths questions
Every Edexcel A-Level Maths exam question on P.7 Differentiation, 2019–2025, with its mark scheme and examiner’s report.
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![June 2024 Pure 1, question 3: f(x) = 1 3 tan 2 2 π x x π x < < Given that the equation f(x) = 0 has a single root α (a) show that α lies in the interval [3.6, 3.7] (2) (b) Find f′(](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/9ma0-01-jun24-03-qp.webp)
All 64 questions on Differentiation, as a list
- June 2025, Pure 1, Q7 (7 marks): O 2 –1 –75 5 C x y Figure 1 Figure 1 shows a sketch of a curve C with equation y = f(x), where f(x) is a quartic expression in x. The curve • has maximum turnin
- June 2025, Pure 1, Q9 (9 marks): O T t v Figure 2 A racing car is driven along a straight road. Figure 2 shows a graph of the speed of the car as it travels along the road. The car starts from
- June 2025, Pure 1, Q15 (9 marks): O D C B A Figure 4 Figure 4 shows the plan view for the design of a stage. The shape of this design consists of a sector of a circle AOB joined to a rectangle O
- June 2025, Pure 2, Q7 (4 marks): Given that 3 8 6 11 1 3 1 3 3 2 x x x x x Ax B C x D x x ∈ x ≠ –1, 3 find the value of each of the constants A, B, C and D. (4)
- June 2025, Pure 2, Q10 (9 marks): Water flows at a constant rate into a large container. There is a tap at the bottom of the container. At time t hours after the tap was opened • the volume of w
- June 2025, Pure 2, Q13 (3 marks): Given that for all values of k, where 0 < ½k½ < 1, the equation sin (nx) = k n ∈ has exactly 6 solutions in the interval 0 x < 2π (i) deduce the value of n
- June 2025, AS Pure, Q6 (7 marks): f (x) = ax 3 + bx 2 + 18x + 9 where a and b are constants. Given that (x + 3) is a factor of f (x) (a) show that –3a + b = 5 (2) Given also that f′ (2) = 14 (b)
- June 2025, AS Pure, Q11 (13 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. r cm l cm Figure 4 Figure 4 s
- June 2025, AS Pure, Q12 (5 marks): y x O C Figure 5 Figure 5 shows a sketch of the curve C. The curve has a minimum turning point on the y-axis and a maximum turning point in quadrant 2. (a) On D
- June 2024, Pure 1, Q3 (6 marks): f(x) = 1 3 tan 2 2 π x x π x < < Given that the equation f(x) = 0 has a single root α (a) show that α lies in the interval [3.6, 3.7] (2) (b) Find f′(
- June 2024, Pure 1, Q4 (3 marks): Given that y = x 2, use differentiation from first principles to show that d d y x = 2x (3)
- June 2024, Pure 1, Q5 (6 marks): The function f is defined by f(x) = 2 2 3 4 x x x (a) Show that f′(x) = ax bx c x 2 2 2 4 where a, b and c are constants to be found. (3) (b) Hence, using a
- June 2024, Pure 1, Q10 (9 marks): A l1 R l2 O x y Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
- June 2024, Pure 2, Q1 (5 marks): y x x x 4 7 5 10 3 2 (a) Find in simplest form (i) d d y x (ii) d d 2 2 y x (3) (b) Hence find the exact value of x when d d 2 2 y x = 0 (2)
- June 2024, Pure 2, Q6 (7 marks): O y = f(x) y = g(x) x y Figure 1 Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where f(x) = e 4x2 – 1 x > 0 g(x) = 8 ln x x > 0 (a)
- June 2024, Pure 2, Q10 (6 marks): O C x y Figure 4 Figure 4 shows a sketch of the curve C with parametric equations x t y t t 3 1 2 1 2 3 The point P with coordinates (4, 2) lies on C. (a
- June 2024, Pure 2, Q15 (12 marks): The curve C has equation ( ) x y x y 3 2 3 3 2 (a) Find an expression for d d y x in terms of x and y. (5) The point P(1, 0) lies on C. (b) Show that the normal
- June 2024, AS Pure, Q8 (10 marks): A B C R O y x l Figure 3 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figu
- June 2024, AS Pure, Q12 (13 marks): x y A B Figure 5 Figure 5 shows the plan view of the design for a swimming pool. The pool is modelled as a quarter of a circle joined to two equal sized rectang
- June 2023, Pure 1, Q4 (5 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve C has equation y =
- June 2023, Pure 1, Q12 (5 marks): y = sin x where x is measured in radians. Use differentiation from first principles to show that d d y x = cos x You may • use without proof the formula for sin
- June 2023, Pure 1, Q15 (13 marks): A curve has equation y = f(x), where f(x) = 7 2 3 x x x e e − x > ln 2 3 (a) Show that f′(x) = 7 2 2 3 3 3 2 e e 2(e x x x x Ax B ( ( ) ) ) where A and B are co
- June 2023, Pure 2, Q1 (4 marks): f(x) = x 3 + 2x 2 – 8x + 5 (a) Find f ′′(x) (2) (b) (i) Solve f ′′(x) = 0 (ii) Hence find the range of values of x for which f(x) is concave. (2)
- June 2023, Pure 2, Q4 (4 marks): Coffee is poured into a cup. The temperature of the coffee, H °C, t minutes after being poured into the cup is modelled by the equation H = Ae – Bt + 30 where A
- June 2023, Pure 2, Q5 (5 marks): The curve C has equation y = f (x) The curve • passes through the point P(3, –10) • has a turning point at P Given that d d y x x x x k 2 9 5 3 2 where k is a c
- June 2023, Pure 2, Q7 (7 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation x 3 + 2xy + 3y 2
- June 2023, Pure 2, Q9 (7 marks): The curve C has parametric equations x = t 2 + 6t – 16 y = 6 ln(t + 3) t > –3 (a) Show that a Cartesian equation for C is y = A ln (x + B) x > –B where A and B
- June 2023, AS Pure, Q1 (6 marks): A curve has equation (a) Find d d y x writing your answer in simplest form. (2) (b) Hence find the range of values of x for which y is decreasing. (4)
- June 2023, AS Pure, Q11 (6 marks): The height, h metres, of a plant, t years after it was first measured, is modelled by the equation Using the model, (a) find the height of the plant when it was
- June 2022, Pure 1, Q6 (6 marks): (2, 8) C x 6 O y Figure 1 Figure 1 shows a sketch of a curve C with equation y = f (x) where f (x) is a cubic expression in x. The curve • passes through the or
- June 2022, Pure 1, Q8 (8 marks): v t T O Figure 2 A car stops at two sets of traffic lights. Figure 2 shows a graph of the speed of the car, v ms−1, as it travels between the two sets of traffi
- June 2022, Pure 1, Q10 (8 marks): A scientist is studying the number of bees and the number of wasps on an island. The number of bees, measured in thousands, Nb , is modelled by the equation Nb
- June 2022, Pure 1, Q15 (10 marks): A 0.8 rad h r r F E D C B Figure 5 A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is mo
- June 2022, Pure 2, Q4 (3 marks): Given that y = 2x 2 use differentiation from first principles to show that d d y x = 4x (3)
- June 2022, Pure 2, Q6 (7 marks): y x O α P Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = 8 sin 1 2 x − 3x + 9 x > 0 and x is measured in ra
- June 2022, Pure 2, Q12 (6 marks): The function f is defined by f(x) = e3 2 4 x x k + where k is a positive constant. (a) Show that fʹ(x) = (12x 2 − 8x + 3k) g(x) where g(x) is a function to be f
- June 2022, Pure 2, Q16 (12 marks): y l P C O x Figure 6 Figure 6 shows a sketch of the curve C with parametric equations x = 2 tan t + 1 y = 2 sec 2 t + 3 − 4 π t 3 π The line l is the normal
- June 2022, AS Pure, Q10 (10 marks): P C O R y x l Figure 2 In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 2 shows
- June 2022, AS Pure, Q12 (12 marks): A company makes drinks containers out of metal. The containers are modelled as closed cylinders with base radius r cm and height h cm and the capacity of each c
- November 2021, Pure 1, Q4 (9 marks): The curve with equation y = f (x) where f (x) = x2 + ln (2x2 − 4x + 5) has a single turning point at x = α (a) Show that α is a solution of the equation 2x3 − 4
- November 2021, Pure 1, Q14 (4 marks): Given that y = x x − + 4 2 x > 0 show that d d y x = A x 1 x > 0 where A is a constant to be found. (4)
- November 2021, Pure 2, Q5 (7 marks): The curve C has equation y = 5x4 – 24x3 + 42x2 – 32x + 11 x ∈ (a) Find (i) d d y x (ii) d d 2 2 y x (3) (b) (i) Verify that C has a stationary point at x = 1
- November 2021, Pure 2, Q7 (9 marks): In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. x C O l R P(5, –13) y Figur
- November 2021, Pure 2, Q8 (9 marks): The curve C has equation px3 + qxy + 3y2 = 26 where p and q are constants. (a) Show that d d y x apx bqy qx cy 2 where a, b and c are integers to be found. (4)
- November 2021, Pure 2, Q13 (6 marks): The curve C has parametric equations x = sin 2θ y = cosec3 θ 0 < θ < π 2 (a) Find an expression for d d y x in terms of θ (3) (b) Hence find the exact value of
- November 2021, AS Pure, Q5 (6 marks): O Q P(2, 10) y x Figure 1 Figure 1 shows part of the curve with equation y = 3x 2 – 2 The point P(2, 10) lies on the curve. (a) Find the gradient of the tangent
- November 2021, AS Pure, Q16 (11 marks): The curve C has equation y = f(x) where f(x) = ax 3 + 15x 2 – 39x + b and a and b are constants. Given • the point (2, 10) lies on C • the gradient of the curve
- November 2020, Pure 1, Q9 (9 marks): y x O C Figure 2 Figure 2 shows a sketch of the curve C with equation y = f (x) where f (x) = 4(x2 − 2)e−2x x (a) Show that f ʹ(x) = 8(2 + x − x2)e−2x (3) (
- November 2020, Pure 1, Q14 (10 marks): A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm3 The volume of the balloon is modelled as decreasing at a
- November 2020, Pure 1, Q15 (7 marks): The curve C has equation x2 tan y = 9 0 < y < 2 π (a) Show that d d y x = − + 18 81 4 x x (4) (b) Prove that C has a point of inflection at x = 27 4 (3)
- November 2020, Pure 2, Q1 (5 marks): 1 The table below shows corresponding values of x and y for y = x x 1 + The values of y are given to 4 significant figures. x 0.5 1 1.5 2 2.5 y 0.5774 0.7071 0.
- November 2020, Pure 2, Q2 (3 marks): Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively. Given that ● P, Q and R lie on a straight line ● Q lies one third o
- November 2020, Pure 2, Q7 (10 marks): y P O x Figure 1 Figure 1 shows a sketch of the curve C with equation y = 4 2 2x x x + − 4 ln x x > 0 (a) Show that d d y x = 12 16 4 2x x x x x + − (4) The poi
- November 2020, Pure 2, Q13 (6 marks): The function g is defined by g (x) = 3 7 2 ln( ) ln( ) x x − − x > 0 x ≠ k where k is a constant. (a) Deduce the value of k. (1) (b) Prove that gʹ (x) > 0 for a
- November 2020, AS Pure, Q1 (5 marks): A curve has equation y = 2x3 – 4x + 5 Find the equation of the tangent to the curve at the point P(2, 13). Write your answer in the form y = mx + c, where m and
- November 2020, AS Pure, Q14 (9 marks): A curve has equation y = g(x). Given that ● g(x) is a cubic expression in which the coefficient of x3 is equal to the coefficient of x ● the curve with equation
- June 2019, Pure 1, Q3 (5 marks): y x x x x = + + ≠− 5 10 1 1 2 2 ( ) (a) Show that d d y x A x n = + ( )1 where A and n are constants to be found. (4) (b) Hence deduce the range of values for x
- June 2019, Pure 1, Q12 (10 marks): f(x) = 10e±x sin x, x . 0 (a) Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4 (4) y O
- June 2019, Pure 1, Q14 (7 marks): The curve C, in the standard Cartesian plane, is defined by the equation x = 4 sin 2\ 4 4 π π y − < < The curve C passes through the origin O (a) Find th
- June 2019, Pure 2, Q11 (11 marks): y x C O Figure 8 Figure 8 shows a sketch of the curve C with equation y = x x , x 0 (a) Find, by firstly taking logarithms, the x coordinate of the turning poin
- June 2019, Pure 2, Q13 (10 marks): r m r m h m Figure 9 [A sphere of radius r has volume 4 3 ʌU࣠3 and surface area 4ʌU࣠2] A manufacturer produces a storage tank. The tank is modelled in the shape
- June 2019, AS Pure, Q5 (5 marks): A curve has equation y = 3x2 + 24 x + 2 x 0 (a) Find, in simplest form, d d y x (3) (b) Hence find the exact range of values of x for which the curve is increas
- June 2019, AS Pure, Q13 (7 marks): y x O k R Figure 3 Figure 3 shows a sketch of part of the curve with equation y = 2x3 – 17x2 + 40x The curve has a minimum turning point at x = k. The region R,
- June 2019, AS Pure, Q14 (9 marks): The value of a car, £V, can be modelled by the equation V = 15 700e– 0.25t + 2300 t \, t . 0 where the age of the car is t years. Using the model, (a) find th
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