Differentiation: AQA A‑Level Maths questions

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

Find questions426

PaperExamSort

426 questions

Q1
June 2025 · Paper 1
1 markG Differentiation
The whole paper ↗
Q11
June 2025 · Paper 1
7 marksG Differentiation
The whole paper ↗
Q13
June 2025 · Paper 1
9 marksG Differentiation C Coordinate geometry in the (x, y) plane
The whole paper ↗
Q15
June 2025 · Paper 1
12 marksI Numerical methods G Differentiation
The whole paper ↗
Q7
June 2025 · Paper 2
9 marksG Differentiation
The whole paper ↗
Q10
June 2025 · Paper 2
13 marksG Differentiation
The whole paper ↗
Q3
June 2025 · Paper 3
1 markG Differentiation F Exponentials and logarithms
The whole paper ↗
Q11
June 2025 · Paper 3
10 marksG Differentiation
The whole paper ↗
Q6
June 2024 · Paper 1
2 marksG Differentiation
The whole paper ↗
Q19
June 2024 · Paper 1
7 marksG Differentiation
The whole paper ↗
Q20
June 2024 · Paper 1
10 marksH Integration G Differentiation
The whole paper ↗
Q5
June 2024 · Paper 2
3 marksG Differentiation
The whole paper ↗
Q10
June 2024 · Paper 2
4 marksG Differentiation
The whole paper ↗
Q4
June 2024 · Paper 3
2 marksG Differentiation F Exponentials and logarithms
The whole paper ↗
Q10
June 2024 · Paper 3
5 marksG Differentiation
The whole paper ↗
Q2
June 2023 · Paper 1
1 markG Differentiation
The whole paper ↗
Q15
June 2023 · Paper 1
9 marksG Differentiation
The whole paper ↗
Q16
June 2023 · Paper 1
14 marksH Integration B Algebra and functions
The whole paper ↗
Q4
June 2023 · Paper 2
7 marksG Differentiation
The whole paper ↗
Q3
June 2023 · Paper 3
1 markG Differentiation
The whole paper ↗

All 65 questions on Differentiation, as a list

June 2025, Paper 1, Q1 (1 mark): The equation of a curve is given by y = 3ex Find an expression for dy dx Circle your answer. [1 mark] 3e–x 3e1 3ex 3xex–1
June 2025, Paper 1, Q11 (7 marks): The equation of a curve is x2y + 4y3 = 8x The curve has two stationary points. 11 (a) Use implicit differentiation to show that at the stationary points y = 4 x
June 2025, Paper 1, Q13 (9 marks): A curve C has parametric equations x = 4(4t + 1)2 y = e–4t for – 1 4 ≤ t ≤ 0 13 (a) Find dy dx in terms of t [3 marks] 13 (b) Find an equation of the tangent to
June 2025, Paper 1, Q15 (12 marks): A curve has equation y = x2 The point Q has coordinates (3, 2.5) The point P on the curve which is closest to the point Q is shown on the diagram below. x Q (3,
June 2025, Paper 2, Q7 (9 marks): The point A lies on the curve with equation y = x3 + px2 + qx + 12 7 (a) Given that A has coordinates (–5, 37), show that 5p – q = 30 [2 marks] 7 (b) Given that
June 2025, Paper 2, Q10 (13 marks): A curve C has equation y = xk ln x for x > 0 where k is a positive integer. 10 (a) Show that dy dx = xk–1 [A + k ln x] where A is a constant to be found. [4 mar
June 2025, Paper 3, Q3 (1 mark): The function f is defined by f(x) = ex for x ∈ ℝ Identify which one of the following statements describes the function f Tick (✓) one box. [1 mark] Decreasing a
June 2025, Paper 3, Q11 (10 marks): A block of ice is melting. At time t minutes, the block is in the shape of a cuboid with dimensions of 4x, 2x and x, as shown in the diagram. x 2 x 4 x All meas
June 2024, Paper 1, Q6 (2 marks): Use the chain rule to find dy dx when y = (x3 + 5x)7 [2 marks]
June 2024, Paper 1, Q19 (7 marks): A curve has equation y3e2x + 2y – 16x = k where k is a constant. The curve has a stationary point on the y-axis. Determine the value of k [7 marks]
June 2024, Paper 1, Q20 (10 marks): A gardener stores rainwater in a cylindrical container. The container has a height of 130 centimetres. The gardener empties the water from the container through
June 2024, Paper 2, Q5 (3 marks): Given that y = x3 sin x find dy dx [3 marks]
June 2024, Paper 2, Q10 (4 marks): The function f is defined by f (x) = x2 + 2 cos x for – π ≤ x ≤ π Determine whether the curve with equation y = f (x) has a point of inflection at the point whe
June 2024, Paper 3, Q4 (2 marks): A curve has equation y = x4 + 2x Find an expression for dy dx [2 marks]
June 2024, Paper 3, Q10 (5 marks): It is given that f (x) = 5x3 + x Use differentiation from first principles to prove that f ' (x) = 15x2 + 1 [5 marks]
June 2023, Paper 1, Q2 (1 mark): Given that y = 2x3 find dy dx Circle your answer. [1 mark] dy dx = 5x2 dy dx = 6x2 dy dx = x4 2 dy dx = 6x3
June 2023, Paper 1, Q15 (9 marks): The curve with equation x2 + 2y3 − 4xy = 0 has a single stationary point at P as shown in the diagram below. O P y x 15 (a) Show that the y-coordinate of P sati
June 2023, Paper 1, Q16 (14 marks): (a) Given that 1 16 − 9x2 × A 4 − 3x + B 4 + 3x find the values of A and B [3 marks] 16 (b) An empty container, in the shape of a cuboid, has length 1.6 metres,
June 2023, Paper 2, Q4 (7 marks): A curve has equation y = x2 8 + 4 x p 4 (a) Find an expression for dy dx [3 marks] 4 (b) The point P with coordinates (4, 10) lies on the curve. Find an equatio
June 2023, Paper 3, Q3 (1 mark): A curve with equation y = f (x) passes through the point (3, 7) Given that f ’(3) = 0 find the equation of the normal to the curve at (3, 7) Circle your answer.
June 2023, Paper 3, Q5 (3 marks): A curve has equation y = 3e2x Find the gradient of the curve at the point where y = 10 [3 marks]
June 2023, Paper 3, Q7 (14 marks): A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below. θ O P R S T Q The po
June 2023, Paper 3, Q9 (12 marks): A water slide is the shape of a curve PQ as shown in Figure 1 below. Figure 1 O P Q The curve can be modelled by the parametric equations x = t − 1 t + 4:8 y =
June 2022, Paper 1, Q4 (1 mark): The graph of y = f (x) where f (x) = ax2 + bx + c is shown in Figure 1. Figure 1 x y Which of the following shows the graph of y = f 0(x) ? Tick (3) one box. [1
June 2022, Paper 1, Q5 (3 marks): Find an equation of the tangent to the curve y = (x × 2)4 at the point where x = 0 [3 marks]
June 2022, Paper 1, Q13 (9 marks): Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor. The entrance to the cave joins th
June 2022, Paper 1, Q15 (16 marks): (a) Given that y = cosecy 15 (a) (i) Express y in terms of siny. [1 mark] 15 (a) (ii) Hence, prove that dy dy = × cosecy cot y [3 marks] 15 (a) (iii) Show that
June 2022, Paper 2, Q2 (1 mark): State the value of lim h ! 0 sin (p + h) − sinp h Circle your answer. [1 mark] cos h −1 0 1
June 2022, Paper 2, Q3 (1 mark): The function f is concave and is represented by one of the graphs below. Identify the graph which represents f. Tick (3) one box. [1 mark] x y O x y O x y O x y
June 2022, Paper 2, Q7 (9 marks): The curve y = 15 − x2 and the isosceles triangle OPQ are shown on the diagram below. x q y O P Q Vertices P and Q lie on the curve such that Q lies vertically a
June 2022, Paper 2, Q17 (7 marks): A particle is moving such that its position vector, r metres, at time t seconds, is given by r = et cos t i + et sin t j Show that the magnitude of the accelera
June 2022, Paper 3, Q6 (9 marks): A design for a surfboard is shown in Figure 1. Figure 1 length width The curve of the top half of the surfboard can be modelled by the parametric equations x =
June 2022, Paper 3, Q8 (7 marks): Water is poured into an empty cone at a constant rate of 8 cm3/s After t seconds the depth of the water in the inverted cone is h cm, as shown in the diagram be
June 2022, Paper 3, Q10 (13 marks): The function f is defined by f (x) = x2 + 10 2x + 5 where f has its maximum possible domain. The curve y = f (x) intersects the line y = x at the points P and Q
November 2021, Paper 1, Q2 (1 mark): Given that y = ln (5x) find dy dx Circle your answer. [1 mark] dy dx = 1 x dy dx = 1 5x dy dx = 5 x dy dx = ln 5
November 2021, Paper 1, Q10 (8 marks): (a) Given that y = tanx use the quotient rule to show that dy dx = sec2 x [3 marks] 10 (b) The region enclosed by the curve y = tan2 x and the horizontal line,
November 2021, Paper 1, Q12 (8 marks): The equation of a curve is (x + y)2 = 4y + 2x + 8 The curve intersects the positive x-axis at the point P. 12 (a) Show that the gradient of the curve at P is −
November 2021, Paper 1, Q14 (10 marks): The curve C is defined for t × 0 by the parametric equations x = t2 + t and y = 4t2 − t3 C is shown in the diagram below. x y O 14 (a) Find the gradient of C at
November 2021, Paper 2, Q2 (1 mark): A curve has equation y = f (x) The curve has a point of inflection at x = 7 It is given that f 0(7) = a and f 00(7) = b , where a and b are real numbers. Identi
November 2021, Paper 2, Q10 (11 marks): The function h is defined by h(x) = x p x − 3 where h has its maximum possible domain. 10 (a) Find the domain of h. Give your answer using set notation. [3 mark
November 2021, Paper 3, Q3 (1 mark): f (x) = 3x2 Obtain lim h ! 0 f (x + h) − f (x) h Circle your answer. [1 mark] 3h2 h x3 3(x + h)2 − 3x2 h 6x for the next question
November 2021, Paper 3, Q5 (13 marks): A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost
November 2021, Paper 3, Q9 (9 marks): A function f is defined for all real values of x as f (x) = x4 + 5x3 The function has exactly two stationary points when x = 0 and x = − 15 4 9 (a) (i) Find f 0
November 2020, Paper 1, Q12 (13 marks): A curve C has equation x3 siny + cos y = Ax where A is a constant. C passes through the point P 3 p , p 6 12 (a) Show that A = 2 [2 marks] 12 (b) (i) Show that
November 2020, Paper 1, Q14 (9 marks): The function f is defined by f (x) = 3x x p × 1 where x ≥ 0 14 (a) f (x) = 0 has a single solution at the point x = a By considering a suitable change of sign,
November 2020, Paper 2, Q8 (10 marks): The curve defined by the parametric equations x = t2 and y = 2t − 2 p × t × 2 p is shown in Figure 1 below. Figure 1 x y O 8 (a) Find a Cartesian equation of th
November 2020, Paper 2, Q9 (10 marks): A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height
November 2020, Paper 3, Q6 (7 marks): The graph of y = f (x) is shown below. (2, 6) (0, 2) O (–2, –2) (–1, 0) x y 6 (a) Sketch the graph of y = f (−x) [2 marks] x O y 6 (b) Sketch the graph of y = 2
June 2019, Paper 1, Q2 (1 mark): Given y = ekx , where k is a constant, find dy dx Circle your answer. [1 mark] dy dx = ekx dy dx = kekx dy dx = kxekx−1 dy dx = ekx k
June 2019, Paper 1, Q10 (4 marks): The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, r, of the bubble is inversely proportional to r
June 2019, Paper 1, Q11 (4 marks): Jodie is attempting to use differentiation from first principles to prove that the gradient of y = sinx is zero when x = p 2 Jodie’s teacher tells her that she
June 2019, Paper 1, Q13 (7 marks): A curve, C, has equation y = e3x−5 x2 Show that C has exactly one stationary point. Fully justify your answer. [7 marks]
June 2019, Paper 1, Q15 (10 marks): (a) At time t hours after a high tide, the height, h metres, of the tide and the velocity, v knots, of the tidal flow can be modelled using the parametric equat
June 2019, Paper 1, Q16 (16 marks): (a) y = e−x( sinx + cos x) Find dy dx Simplify your answer. [3 marks] 16 (b) Hence, show that ( e−x sinx dx = ae−x( sinx + cos x) + c where a is a rational numb
June 2019, Paper 2, Q7 (10 marks): (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of x3 [2 marks] x y 7 (b) The function f (x) is de
June 2019, Paper 2, Q16 (16 marks): An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete’s velocity, v m s−1, may be modelled by v = 11:71 − 11:68e−0
June 2019, Paper 3, Q9 (15 marks): A curve has equation x2y2 + xy4 = 12 9 (a) Prove that the curve does not intersect the coordinate axes. [2 marks] 9 (b) (i) Show that dy dx = − 2xy + y3 2x2 + 4
June 2018, Paper 1, Q1 (1 mark): y = 1 x2 Find an expression for dy dx Circle your answer. [1 mark] dy dx = 0 2x dy dx = x−2 dy dx = − 2 x dy dx = − 2 x3
June 2018, Paper 1, Q5 (6 marks): A curve is defined by the parametric equations x = 4 × 2−t + 3 y = 3 × 2t − 5 5 (a) Show that dy dx = −3 4 × 22t [3 marks] 5 (b) Find the Cartesian equation of
June 2018, Paper 1, Q13 (10 marks): A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company mode
June 2018, Paper 1, Q15 (6 marks): A curve has equation y = x3 − 48x The point A on the curve has x coordinate −4 The point B on the curve has x coordinate −4 + h 15 (a) Show that the gradient of
June 2018, Paper 2, Q6 (7 marks): Find the coordinates of the stationary point of the curve with equation (x + y − 2)2 = ey − 1 [7 marks]
June 2018, Paper 2, Q9 (14 marks): A market trader notices that daily sales are dependent on two variables: number of hours, t, after the stall opens total sales, x, in pounds since the stall ope
June 2018, Paper 3, Q2 (1 mark): A curve has equation y = x5 + 4x3 + 7x + q where q is a positive constant. Find the gradient of the curve at the point where x = 0 Circle your answer. [1 mark]
June 2018, Paper 3, Q6 (13 marks): A function f is defined by f (x) = x 2x − 2 p 6 (a) State the maximum possible domain of f. [2 marks] 6 (b) Use the quotient rule to show that f 0(x) = x − 2 (2

Questions, mark schemes and examiner’s reports are © AQA, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every AQA A-Level Maths paper.

Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.