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Differentiation II · CAPE Pure Mathematics Unit 2

92 past-paper questions on Differentiation II, part of Complex Numbers and Calculus II, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Differentiation II
  1. 1(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Copy the diagram and on the same axes, sketch the graph of g(x) = ln x.
  2. 1(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Describe clearly the relationship between f(x) = e^x and g(x) = ln x.
  3. 1(a)(iii)a)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Using a calculator, find the value of r.
  4. 1(a)(iii)b)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Using a calculator, find the value of p.
  5. 1(c)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Find the values of x in R for which e^x + 3e^(-x) = 4.
  6. 2(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Find dy/dx in terms of t.
  7. 2(a)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Find the gradient of the normal to the curve at the point t = 2.
  8. 4(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Show that f is everywhere strictly decreasing.
  9. 1(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Show that f'(x) = x^2 ln x (3 ln x + 2).
  10. 1(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Show that f''(x) = 6x ln^2 x + 10x ln x + 2x.
  11. 1(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Copy and complete the table for values of 2^x and e^{-x} using a calculator, approximating values to 2 decimal places.
  12. 1(b)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2On the same pair of axes using a scale of 4 cm for 1 unit on the x-axis and 4 cm for 1 unit on the y-axis, draw the graphs of y = 2^x and y = e^{-x} for -1 \le x \le 3, x \in \mathbb{R}.
  13. 2(b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find \frac{dy}{dx} when y = \tan^n x.
  14. 1(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Differentiate with respect to x: e^{4x} \cos(\pi x)
  15. 1(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Differentiate with respect to x: \ln\left(\frac{x^2 + 1}{\sqrt{x}}\right)
  16. 1(b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Given y = 3^{-x}, show, by using logarithms, that \frac{\mathrm{d}y}{\mathrm{d}x} = -3^{-x} \ln 3.
  17. 1(b)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Given that u = e^{2x} + e^{-2x} and v = e^{2x} - e^{-2x}, show that…
  18. 1(c)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Differentiate (x \ln x) \sin^{-1} 2x with respect to x.
  19. 1(c)(ii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that \frac{\mathrm{d}y}{\mathrm{d}x} = t + \frac{1}{t}.
  20. 1(c)(ii)b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that C has points of inflexion at (8, 8) and (8, -8).
  21. 1(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find \frac{\mathrm{d}y}{\mathrm{d}x} if y = \sin^2 5x + \sin^2 3x + \cos^2 3x.
  22. 1(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find \frac{\mathrm{d}y}{\mathrm{d}x} if y = \sqrt{\cos x^2}.
  23. 1(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find \frac{\mathrm{d}y}{\mathrm{d}x} if y = x^x.
  24. 1(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Given that y = \cos^{-1} x, where 0 \le \cos^{-1} x \le \pi, prove that \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{1}{\sqrt{1 - x^2}}.
  25. 1(b)(ii)a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Show that \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\sqrt{1 + t}}{2}.
  26. 1(b)(ii)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence, find \frac{\mathrm{d}^2y}{\mathrm{d}x^2} in terms of t, giving your answer in simplified form.
  27. 1(b)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Given that y = e^{\tan^{-1}(2x)}, where -\frac{1}{2}\pi < \tan^{-1}(2x) < \frac{1}{2}\pi, show that (1 + 4x^2)\frac{\mathrm{d}y}{\mathrm{d}x} = 2y.
  28. 1(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Hence, show that (1 + 4x^2)^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 4y(1 - 4x).
  29. 2(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Given that n is a positive integer, find \frac{\mathrm{d}}{\mathrm{d}x}[x(\ln x)^n].
  30. 1(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find \frac{dy}{dx} if x^2 + y^2 - 2x + 2y - 14 = 0.
  31. 1(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find \frac{dy}{dx} if y = e^{\cos x}.
  32. 1(a)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find \frac{dy}{dx} if y = \cos^2 6x + \sin^2 8x.
  33. 1(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that x \frac{dy}{dx} = y - \cos \left(\frac{1}{x}\right).
  34. 1(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that x^4 \frac{d^2y}{dx^2} + y = 0.
  35. 1(c)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the gradient of the tangent to the curve at the point where t = 4.
  36. 1(c)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the equation of the tangent to the curve at the point where t = 4.
  37. 1(a)(i)a)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find \frac{\mathrm{d}y}{\mathrm{d}x} and \frac{\mathrm{d}^2y}{\mathrm{d}x^2}.
  38. 1(a)(i)b)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the x-coordinates of the points at which \frac{\mathrm{d}y}{\mathrm{d}x} = 0.
  39. 1(a)(i)c)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the x-coordinates of the points at which \frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 0.
  40. 1(a)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, determine if the coordinates identified in (i) b) and c) above are at the maxima, minima or points of inflection of y = x^2 e^x.
  41. 1(b)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the gradient of a tangent to the curve at the point with parameter t.
  42. 1(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the equation of the tangent at the point where t = \frac{1}{2}.
  43. 1(a)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Calculate the gradient of the curve \ln(x^2 y) - \sin y = 3x - 2y at the point (1, 0).
  44. 1(b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Let f(x, y, z) = 3yz^2 - e^{4x}\cos 4z - 3y^2 - 4 = 0. Given that \frac{\partial z}{\partial y} = -\frac{\partial f / \partial y}{\partial f / \partial z}, determine \frac{\partial z}{\partial y} in terms of x,…
  45. 1(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Differentiate, with respect to x, y = \ln(x^2 + 4) - x \tan^{-1}\left(\frac{x}{2}\right).
  46. 1(a)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2A curve is defined parametrically as x = a\cos^3 t, y = a\sin^3 t. Show that the tangent at the point P(x, y) is the line y\cos t + x\sin t = a\sin t\cos t.
  47. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If f(x) = \ln 2x, then f'(x) =
  48. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The number of bacteria present in a culture is modelled by y = y_0 e^{kt}, where k > 0, y is the population after t hours, and y_0 is the initial population. The rate of growth, c, when t = 5 is given by
  49. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1\frac{d}{dx}(\ln x)^3 =
  50. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1A curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The simplest expression for the gradient of the tangent in terms of t is
  51. 1(c)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Show that \frac{dy}{dx} = \frac{e^t (1 - t^2)}{t^2 + t - 1}.
  52. 1(c)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, show that f has no stationary value.
  53. 2(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use implicit differentiation to show that \frac{dy}{dx} = -\frac{8x + 3y^2 + 7}{3(1 + 2xy)}.
  54. 2(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Show that for f(x, y) = 4x^2 + 3xy^2 + 7x + 3y,…
  55. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If x^2y - xy^2 = 10, then \frac{dy}{dx} is equal to
  56. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The derivative of \ln x^{\frac{1}{3}} is
  57. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If \frac{dy}{dx} = 2xy, then the value of \frac{d^2y}{dx^2} at the point (1, 2) is
  58. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1A curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The expression for \frac{dy}{dx} is given by
  59. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If f(x, y) is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] and \frac{\partial f}{\partial y} = -e^x \sin(x + y), then which of the following is TRUE?
  60. 1(c)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Determine the x-coordinates of the two stationary values of f.
  61. 2(a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Determine partial derivative of w with respect to x.
  62. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that y = \tan^{-1}(2x), then \frac{dx}{dy} equals
  63. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given e^{x+y} - x = 0, then \frac{dy}{dx} is equal to
  64. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The number of bacteria present in a culture is modelled by y = y_0 e^{kt}, where k > 0, y is the population after t hours, and y_0 is the initial population. The rate of growth, c, when t = 5 is given by
  65. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The derivative of the function y = \ln\left(\frac{\cos x}{\sin x}\right) with respect to x is
  66. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that f(x) = \ln 3x^2, then f'(-2) equals
  67. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1If f is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] and \frac{\partial f}{\partial y} = -e^x \sin(x + y), then which of the following is TRUE?
  68. 1(a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Find the first derivative of the function f(x) = \cos^{-1}(\sin^{-1} x).
  69. 1(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Given that \frac{\partial w}{\partial x} = -\frac{1}{9} at the point (4, y_0), calculate the value of y_0.
  70. 1(b)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Show that \frac{\partial^2 w}{\partial y \partial x} - 2 \frac{\partial^2 w}{\partial y^2} = 0.
  71. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The derivative of \ln\left(\frac{1}{\sqrt[3]{x}}\right) is
  72. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1If f(x, y) is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] \text{ and} \frac{\partial f}{\partial y} = -e^x \sin(x + y), \text{ then which of the} following is true?
  73. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1If \frac{dy}{dx} = 2xy, then the value of \frac{d^2y}{dx^2} at the point (1, 2) is
  74. 1(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Determine the gradient of the curve at the point (1/2, 1/2).
  75. 1(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Hence, or otherwise, determine the x and y intercepts of the tangent to the curve at the point (1/2, 1/2).
  76. 1(b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Let the function f(x, y) = sin(kx) sin(aky). Determine d^2 f(x, y) / (dx dy).
  77. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that y = \tan^{-1}(2x), then \frac{dx}{dy} equals
  78. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The derivative of the function y = \ln\left(\frac{\cos x}{\sin x}\right) with respect to x is
  79. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1If f(x) = \ln 2x, then f''(x) =
  80. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1If f is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] and \frac{\partial f}{\partial y} = -e^x\sin(x + y), then which of the following is true?
  81. 1(a)6 marks· Pure Mathematics · Unit 2 Q1 1(a)Given that 3y² - 4xy + sin xy = 5, determine dy/dx.
  82. 1(a)(i)5 marks· Pure Mathematics · Unit 2 Q1 1(a)(i)Use implicit differentiation to show that dy/dx = -(8x + 3y² + 7) / (3(1 + 2xy)).
  83. 1(a)(ii)5 marks· Pure Mathematics · Unit 2 Q1 1(a)(ii)Show that 6(∂f(x,y)/∂y) - 10 = (∂²f(x,y)/∂y²) + (∂²f(x,y)/∂y∂x) + (∂²f(x,y)/∂x²).
  84. 1(b)3 marks· Pure Mathematics · Unit 2 Q1 1(b)Differentiate cos⁻¹ (3x - 2), expressing your answer in its simplest form.
  85. 1(b)5 marks· Pure Mathematics · Unit 2 Q1 1(b)Given that x²y – 2xy² = cos(xy), determine dy/dx.
  86. 1(c)6 marks· Pure Mathematics · Unit 2 Q1 1(c)Show that the derivative of sin⁻¹(cos x / (1 + sin x)) with respect to x is -1 / √( (1+sin x)² - cos² x ).
  87. 1(c)6 marks· Pure Mathematics · Unit 2 Q1 1(c)Determine an expression for dy/dx in terms of x and y.
  88. 1(d)6 marks· Pure Mathematics · Unit 2 Q1 1(d)Given that (x² + y²)³ = ax²y, use implicit differentiation to determine dy/dx.
  89. 1(d)5 marks· Pure Mathematics · Unit 2 Q1 1(d)Given that x = 2t - sin t and y = 1 - cos t, determine d²y/dx².
  90. 1(d)6 marks· Pure Mathematics · Unit 2 Q1 1(d)A curve is defined parametrically by x = (3 – 2t)², y = t³ – 2t. Determine the equation of the tangent to the curve at the point where t = 2.
  91. 1(d)7 marks· Pure Mathematics · Unit 2 Q1 1(d)Show that y = -x + π/2 is the tangent to the curve y = ln(1 + sin 2x) at the point where x = π/2.
  92. 3(c)10 marks· Pure Mathematics · Unit 2 Q3 3(c)Use mathematical induction to prove that dⁿ/dxⁿ (eˣ sin x) = 2ⁿ/² eˣ sin(x + nπ/4). You may use the fact that sin x + cos x = √2 sin(x + π/4) and that dᵏ⁺¹/dxᵏ⁺¹ f(x) = d/dx (dᵏ/dxᵏ f(x)).

Related topics in Complex Numbers and Calculus II