Question Simulator

IGCSE Additional Mathematics (0606)

Differentiation Banker

35 questions · about 13% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 12 · Q54 marksTangents and normals
  • 2026 May/June · Paper 12 · Q85 marksSmall changes and rates of change
  • 2026 May/June · Paper 22 · Q3 (a)3 marksDerivatives (chain, product, quotient)
  • 2026 May/June · Paper 22 · Q124 marksSmall changes and rates of change

2025

  • 2025 Oct/Nov · Paper 12 · Q9 (b)1 markStationary points
  • 2025 Oct/Nov · Paper 12 · Q12 (a, b)6 marksDerivatives (chain, product, quotient) · Small changes and rates of change
  • 2025 Oct/Nov · Paper 22 · Q7 (a)2 marksDerivatives (chain, product, quotient)
  • 2025 May/June · Paper 12 · Q8 (a, b)6 marksDerivatives (chain, product, quotient) · Small changes and rates of change
  • 2025 May/June · Paper 22 · Q4 (a, b)9 marksDerivatives (chain, product, quotient) · Tangents and normals
  • 2025 May/June · Paper 22 · Q66 marksSmall changes and rates of change

2024

  • 2024 Oct/Nov · Paper 12 · Q2 (a)4 marksStationary points
  • 2024 Oct/Nov · Paper 12 · Q5 (b)1 markDerivatives (chain, product, quotient)
  • 2024 Oct/Nov · Paper 12 · Q76 marksSmall changes and rates of change
  • 2024 Oct/Nov · Paper 12 · Q88 marksTangents and normals
  • 2024 Oct/Nov · Paper 22 · Q2 (a, b, c)8 marksDerivatives (chain, product, quotient) · Small changes and rates of change · Stationary points
  • 2024 Oct/Nov · Paper 22 · Q10 (a, c)8 marksStationary points · Tangents and normals
  • 2024 Oct/Nov · Paper 22 · Q126 marksPractical maxima and minima
  • 2024 May/June · Paper 12 · Q34 marksDerivatives (chain, product, quotient)
  • 2024 May/June · Paper 12 · Q8 (a, b)8 marksDerivatives (chain, product, quotient) · Stationary points
  • 2024 May/June · Paper 12 · Q11 (a)2 marksDerivatives (chain, product, quotient)
  • 2024 May/June · Paper 22 · Q56 marksTangents and normals

2023

  • 2023 Oct/Nov · Paper 12 · Q12 (b)5 marksDerivatives (chain, product, quotient)
  • 2023 Oct/Nov · Paper 22 · Q5 (a)5 marksTangents and normals
  • 2023 Oct/Nov · Paper 22 · Q8 (a, b)5 marksDerivatives (chain, product, quotient) · Tangents and normals
  • 2023 May/June · Paper 22 · Q2 (a, b)8 marksStationary points
  • 2023 May/June · Paper 22 · Q46 marksSmall changes and rates of change
  • 2023 May/June · Paper 22 · Q7 (a, b, c(i))9 marksDerivatives (chain, product, quotient) · Small changes and rates of change
  • 2023 May/June · Paper 22 · Q87 marksTangents and normals

2022

  • 2022 Oct/Nov · Paper 22 · Q36 marksTangents and normals
  • 2022 Oct/Nov · Paper 22 · Q8 (a, b, c)10 marksPractical maxima and minima
  • 2022 May/June · Paper 12 · Q6 (c)2 marksDerivatives (chain, product, quotient)
  • 2022 May/June · Paper 12 · Q99 marksTangents and normals
  • 2022 May/June · Paper 22 · Q45 marksSmall changes and rates of change
  • 2022 May/June · Paper 22 · Q74 marksDerivatives (chain, product, quotient)
  • 2022 May/June · Paper 22 · Q8 (a)5 marksStationary points

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Tangents and normals · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Find the equation of the tangent to the curve y = x³ − 4x + 1 at the point where x = 2. [4]
Show answer
  1. (a) y = 8x − 15
    dy/dx = 3x² − 4 B1; at x = 2, gradient 8 B1, y = 1 B1; y − 1 = 8(x − 2), y = 8x − 15 A1
Q2Tangents and normals · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Find the equation of the tangent to the curve y = e^(2x) − 3x at the point where the curve crosses the y-axis. [3]
  2. (b) Write down the equation of the normal at the same point. [1]
Show answer
  1. (a) y = 1 − x
    dy/dx = 2e^(2x) − 3 B1; at x = 0, y = 1 and gradient −1 M1; y = 1 − x A1
  2. (b) y = x + 1
    gradient 1, through (0, 1) B1
Q3Tangents and normals · Core · Paper 1 style · no calculator · 5 marks
  1. (a) Find the equation of the normal to the curve y = ln(3x − 2) at the point where x = 1. [4]
  2. (b) Find the coordinates of the point where the normal meets the y-axis. [1]
Show answer
  1. (a) y = −⅓(x − 1), i.e. x + 3y = 1
    dy/dx = 3/(3x − 2) M1 A1; at x = 1, y = 0, gradient 3 B1; normal gradient −⅓, x + 3y = 1 A1
  2. (b) (0, ⅓)
    B1
Q4Tangents and normals · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Find the equation of the normal to the curve y = 2x² − 3x + 5 at the point (1, 4). [3]
  2. (b) Find the coordinates of the point where this normal meets the curve again. [3]
Show answer
  1. (a) y = −x + 5
    dy/dx = 4x − 3 = 1 at x = 1 M1; normal gradient −1 M1; y = −x + 5 A1
  2. (b) (0, 5)
    2x² − 3x + 5 = −x + 5 M1; 2x(x − 1) = 0 M1; (0, 5) A1
Q5Tangents and normals · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Show that the tangent to the curve y = x sin x at the point where x = π/2 passes through the origin. [4]
Show answer
  1. (a) Tangent is y = x, which passes through (0, 0)
    dy/dx = sin x + x cos x M1 A1; at x = π/2, y = π/2, gradient 1 A1; y − π/2 = x − π/2 ⇒ y = x A1
Q6Tangents and normals · Standard · Paper 2 style · calculator · 6 marks
  1. (a) Find the equation of the normal to the curve y = (x² + 3)/(x − 1) at the point where x = 2. [5]
  2. (b) Find the x-coordinate of the point where this normal meets the x-axis. [1]
Show answer
  1. (a) 3y = x + 19
    quotient rule: dy/dx = [2x(x − 1) − (x² + 3)]/(x − 1)² M1 A1; at x = 2, y = 7, gradient −3 A1; normal gradient ⅓ M1; 3y = x + 19 A1
  2. (b) x = −19
    B1
Q7Tangents and normals · Challenge · Paper 1 style · no calculator · 8 marks
  1. (a) Find the equations of the tangent and the normal to the curve y = √(4x + 1) at the point where x = 2. [5]
  2. (b) The tangent and the normal meet the x-axis at A and B respectively. Find the area of the triangle formed by A, B and the point (2, 3). [3]
Show answer
  1. (a) tangent 3y = 2x + 5; normal 2y = −3x + 12
    dy/dx = 2(4x + 1)^(−½) M1 A1; at x = 2, y = 3, gradient 2/3 A1; tangent y − 3 = ⅔(x − 2) A1; normal y − 3 = −3/2(x − 2) A1
  2. (b) 39/4 (9.75) square units
    A(−2.5, 0), B(4, 0) B1 B1; area = ½ × 6.5 × 3 = 9.75 B1
Q8Tangents and normals · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) The line y = 3x − 4 is a tangent to the curve y = x² + kx, where k is a constant. Find the possible values of k and, for each value, the coordinates of the point of contact. [6]
Show answer
  1. (a) k = −1, contact (2, 2); k = 7, contact (−2, −10)
    2x + k = 3 M1; x² + kx = 3x − 4 M1; substitute k = 3 − 2x: −x² = −4 M1; x = ±2 A1; k = −1 at (2, 2) A1; k = 7 at (−2, −10) A1
Q9Derivatives (chain, product, quotient) · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Differentiate (2x − 5)⁶ with respect to x. [2]
  2. (b) Differentiate x²e^(3x) with respect to x. [2]
Show answer
  1. (a) 12(2x − 5)⁵
    6(2x − 5)⁵ × 2 M1 A1
  2. (b) e^(3x)(2x + 3x²)
    product rule M1; 2xe^(3x) + 3x²e^(3x) A1
Q10Derivatives (chain, product, quotient) · Core · Paper 1 style · no calculator · 5 marks
  1. (a) Find d/dx (sin 2x ÷ x). [3]
  2. (b) Find d/dx (tan 3x). [2]
Show answer
  1. (a) (2x cos 2x − sin 2x)/x²
    quotient rule M1; 2 cos 2x seen B1; correct A1
  2. (b) 3 sec² 3x
    sec² 3x M1; × 3 A1
Q11Derivatives (chain, product, quotient) · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Given that y = e^(2x) cos x, find dy/dx. [3]
  2. (b) Find the exact value of dy/dx when x = π/2. [1]
Show answer
  1. (a) e^(2x)(2 cos x − sin x)
    product rule M1; 2e^(2x) cos x B1; − e^(2x) sin x A1
  2. (b) −e^π
    e^π(0 − 1) B1
Q12Derivatives (chain, product, quotient) · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Show that d/dx [x ÷ √(x + 1)] = (x + 2) ÷ [2(x + 1)^(3/2)]. [4]
Show answer
  1. (a) shown
    quotient (or product) rule M1; [√(x + 1) − x/(2√(x + 1))] ÷ (x + 1) A1; multiply through by 2√(x + 1) M1; = (2x + 2 − x) ÷ [2(x + 1)^(3/2)] = (x + 2) ÷ [2(x + 1)^(3/2)] A1
Q13Derivatives (chain, product, quotient) · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) Given that y = x ln(2x), find dy/dx and d²y/dx². [3]
  2. (b) Find the exact value of x for which dy/dx = 0. [2]
Show answer
  1. (a) dy/dx = ln(2x) + 1; d²y/dx² = 1/x
    product rule M1; ln 2x + x × (1/x) A1; 1/x A1
  2. (b) x = 1/(2e)
    ln 2x = −1 M1; x = e⁻¹/2 A1
Q14Derivatives (chain, product, quotient) · Standard · Paper 2 style · calculator · 6 marks
  1. (a) Given that y = (3x + 1)/(x − 2), show that dy/dx = −7/(x − 2)². [3]
  2. (b) Explain why the curve has no stationary points. [1]
  3. (c) Find the x-coordinates of the points on the curve where the gradient is −7/4. [2]
Show answer
  1. (a) shown
    quotient rule M1; [3(x − 2) − (3x + 1)]/(x − 2)² A1; = −7/(x − 2)² A1
  2. (b) −7/(x − 2)² is never zero (it is always negative)
    B1
  3. (c) x = 0, x = 4
    (x − 2)² = 4 M1; x = 0 or 4 A1
Q15Derivatives (chain, product, quotient) · Challenge · Paper 1 style · no calculator · 7 marks
  1. (a) Given that y = (x² − 1)√(2x + 3), show that dy/dx = (5x² + 6x − 1)/√(2x + 3). [4]
  2. (b) Hence find the exact x-coordinates of the stationary points of the curve. [3]
Show answer
  1. (a) shown
    product rule M1; 2x√(2x + 3) + (x² − 1)(2x + 3)^(−½) A1; common denominator M1; [2x(2x + 3) + x² − 1]/√(2x + 3) = (5x² + 6x − 1)/√(2x + 3) A1
  2. (b) x = (−3 ± √14)/5
    5x² + 6x − 1 = 0 M1; x = (−6 ± √56)/10 A1; = (−3 ± √14)/5, both > −1.5 so both valid A1
Q16Derivatives (chain, product, quotient) · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Given that y = sin³(2x), find dy/dx. [3]
  2. (b) Find the exact value of dy/dx when x = π/12. [2]
Show answer
  1. (a) 6 sin²(2x) cos(2x)
    chain rule: 3 sin²2x × cos 2x × 2 M1 M1 A1
  2. (b) 3√3/4
    6 × (½)² × (√3/2) M1 = 3√3/4 A1
Q17Small changes and rates of change · Core · Paper 2 style · calculator · 4 marks
  1. (a) A spherical balloon is being inflated so that its radius increases at a constant rate of 0.2 cm per second. Find the rate of increase of the volume of the balloon at the instant when the radius is 5 cm. Give your answer correct to 3 significant figures. [4]
Show answer
  1. (a) 62.8 cm³/s
    V = (4/3)πr³, dV/dr = 4πr² B1; dV/dt = dV/dr × dr/dt M1; = 4π(25)(0.2) M1 = 20π = 62.8 A1
Q18Small changes and rates of change · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Given that y = 4/x², find dy/dx. [1]
  2. (b) Hence find, in terms of h, the approximate change in y when x increases from 2 to 2 + h, where h is small. [2]
Show answer
  1. (a) −8/x³
    B1
  2. (b) −h (a decrease of h)
    dy/dx at x = 2 is −1 M1; δy ≈ −1 × h A1
Q19Small changes and rates of change · Standard · Paper 2 style · calculator · 4 marks
  1. (a) An oil spill in the Arabian Sea forms a circle whose area is increasing at a constant rate of 50 m² per second. Find the rate at which the radius is increasing at the instant when the radius is 20 m. Give your answer correct to 3 significant figures. [4]
Show answer
  1. (a) 0.398 m/s
    A = πr², dA/dr = 2πr B1; dA/dt = dA/dr × dr/dt M1; 50 = 40π × dr/dt M1; dr/dt = 0.398 A1
Q20Small changes and rates of change · Standard · Paper 2 style · calculator · 5 marks

A container is in the shape of a right circular cone with its vertex at the bottom and its axis vertical. The radius of the top of the cone is half its height.

  1. (a) Show that, when the depth of water is h cm, the volume of water is V = πh³/12 cm³. [2]
  2. (b) Water is poured into the container at a constant rate of 3 cm³ per second. Find the rate at which the depth is increasing when h = 6. Give your answer correct to 3 significant figures. [3]
Show answer
  1. (a) shown
    radius of surface = h/2 B1; V = ⅓π(h/2)²h = πh³/12 B1
  2. (b) 0.106 cm/s
    dV/dh = πh²/4 = 9π M1; dh/dt = 3 ÷ 9π M1 = 0.106 A1
Q21Small changes and rates of change · Challenge · Paper 2 style · calculator · 6 marks
  1. (a) The variables x and y are related by y = (3x − 1)/(x + 2). Find dy/dx. [2]
  2. (b) Given that x is increasing at a rate of 0.5 units per second, find the rate of change of y when x = 1. [2]
  3. (c) Use differentiation to find the approximate change in y when x increases from 1 to 1.03. [2]
Show answer
  1. (a) 7/(x + 2)²
    quotient rule M1; [3(x + 2) − (3x − 1)]/(x + 2)² = 7/(x + 2)² A1
  2. (b) 7/18 units per second (0.389)
    dy/dt = (7/9) × 0.5 M1 = 7/18 A1
  3. (c) 0.0233 (an increase)
    δy ≈ (7/9) × 0.03 M1 = 0.0233 A1
Q22Small changes and rates of change · Challenge · Paper 2 style · calculator · 5 marks
  1. (a) The surface area of a cube is increasing at a constant rate of 6 cm² per second. Find the rate at which the length of each edge is increasing when the edge is 4 cm long. [3]
  2. (b) Hence find the rate at which the volume of the cube is increasing at this instant. [2]
Show answer
  1. (a) 0.125 cm/s
    A = 6x², dA/dx = 12x B1; 6 = 48 × dx/dt M1; dx/dt = 0.125 A1
  2. (b) 6 cm³/s
    dV/dt = 3x² × dx/dt M1 = 48 × 0.125 = 6 A1
Q23Stationary points · Core · Paper 1 style · no calculator · 6 marks
  1. (a) Find the coordinates of the stationary points on the curve y = x³ − 6x² + 9x + 2. [4]
  2. (b) Determine the nature of each stationary point. [2]
Show answer
  1. (a) (1, 6) and (3, 2)
    dy/dx = 3x² − 12x + 9 B1; = 0 M1; x = 1, 3 A1; y = 6, 2 A1
  2. (b) (1, 6) maximum; (3, 2) minimum
    d²y/dx² = 6x − 12 M1; −6 < 0 max, 6 > 0 min A1
Q24Stationary points · Core · Paper 1 style · no calculator · 5 marks
  1. (a) Find the coordinates of the stationary point on the curve y = 2x + 8/x², and determine its nature. [5]
Show answer
  1. (a) (2, 6), a minimum
    dy/dx = 2 − 16/x³ B1; = 0 gives x³ = 8, x = 2 M1 A1; y = 6 A1; d²y/dx² = 48/x⁴ > 0, minimum A1
Q25Stationary points · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Find the exact coordinates of the stationary point on the curve y = xe^(−2x). [4]
  2. (b) Determine the nature of this stationary point. [2]
Show answer
  1. (a) (½, 1/(2e))
    product rule M1; dy/dx = e^(−2x)(1 − 2x) A1; x = ½ A1; y = ½e⁻¹ A1
  2. (b) maximum
    e.g. dy/dx > 0 for x < ½ and < 0 for x > ½, or d²y/dx² = e^(−2x)(4x − 4) = −2/e < 0 M1 A1
Q26Stationary points · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Find the exact coordinates of the stationary point on the curve y = (ln x)/x, where x > 0. [4]
  2. (b) Show that this stationary point is a maximum. [2]
Show answer
  1. (a) (e, 1/e)
    quotient rule: dy/dx = (1 − ln x)/x² M1 A1; ln x = 1, x = e A1; y = 1/e A1
  2. (b) d²y/dx² = (2 ln x − 3)/x³ = −1/e³ < 0
    d²y/dx² M1; negative at x = e so maximum A1
Q27Stationary points · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) The curve y = x² + k/x, where k is a constant, has a stationary point at x = 3. Find the value of k. [3]
  2. (b) Find the y-coordinate of this stationary point and determine its nature. [3]
Show answer
  1. (a) k = 54
    dy/dx = 2x − k/x² M1; 6 − k/9 = 0 M1; k = 54 A1
  2. (b) y = 27, minimum
    y = 9 + 18 = 27 B1; d²y/dx² = 2 + 2k/x³ = 6 > 0 M1 minimum A1
Q28Practical maxima and minima · Core · Paper 2 style · calculator · 5 marks

A farmer in Punjab uses 120 m of fencing to enclose a rectangular field next to a straight river. The river forms one side, so fencing is needed on the other three sides only. The two sides perpendicular to the river each have length x m.

  1. (a) Show that the area A m² of the field is given by A = 120x − 2x². [1]
  2. (b) Find the value of x for which A is a maximum, and find this maximum area. [4]
Show answer
  1. (a) shown
    length parallel to river = 120 − 2x, so A = x(120 − 2x) B1
  2. (b) x = 30, A = 1800 m²
    dA/dx = 120 − 4x M1; = 0, x = 30 A1; A = 1800 A1; d²A/dx² = −4 < 0, maximum B1
Q29Practical maxima and minima · Standard · Paper 2 style · calculator · 7 marks

A closed box is a cuboid with a base x cm by 2x cm and height h cm. Its volume is 576 cm³.

  1. (a) Show that the total surface area, A cm², of the box is given by A = 4x² + 1728/x. [3]
  2. (b) Find the value of x for which A is a minimum, and find this minimum surface area. [4]
Show answer
  1. (a) shown
    2x²h = 576 so h = 288/x² B1; A = 2(2x² + xh + 2xh) = 4x² + 6xh M1; = 4x² + 1728/x A1
  2. (b) x = 6, A = 432 cm²
    dA/dx = 8x − 1728/x² M1; x³ = 216, x = 6 A1; A = 144 + 288 = 432 A1; d²A/dx² = 8 + 3456/x³ > 0, minimum B1
Q30Practical maxima and minima · Challenge · Paper 2 style · calculator · 8 marks

A food company designs a closed cylindrical tin of radius r cm and height h cm to hold 500 cm³ of mango pulp.

  1. (a) Show that the total surface area A cm² of the tin is given by A = 2πr² + 1000/r. [2]
  2. (b) Find the value of r for which A is stationary, giving your answer correct to 3 significant figures, and show that this gives a minimum. [4]
  3. (c) Find the minimum surface area and show that at this value the height of the tin equals its diameter. [2]
Show answer
  1. (a) shown
    πr²h = 500 so h = 500/(πr²) B1; A = 2πr² + 2πrh = 2πr² + 1000/r B1
  2. (b) r = 4.30 cm; d²A/dr² = 4π + 2000/r³ > 0, minimum
    dA/dr = 4πr − 1000/r² M1; r³ = 250/π M1; r = 4.30 A1; second derivative positive A1
  3. (c) A = 349 cm²; h = 8.60 = 2r
    A = 348.7 B1; h = 500/(πr²) = 8.60 = 2r B1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Tangents and normals Banker10 questions · 8/9 sessions
Derivatives (chain, product, quotient) Banker15 questions · 8/9 sessions
Small changes and rates of change 10 questions · 6/9 sessions
Stationary points 7 questions · 5/9 sessions
Practical maxima and minima Rare2 questions · 2/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Tangents and normals Banker578/9
Derivatives (chain, product, quotient) Banker518/9
Small changes and rates of change 426/9
Stationary points 275/9
Practical maxima and minima Rare162/9

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