Further Differentiation

Maths, HSSC-Punjab Second Year. This chapter is split into 67 topics and has 1,626 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Optimisation of cubic runtime and level models 26 questions
2. Quotient rule 26 questions
3. Chain rule with trigonometric functions 26 questions
4. Second derivative test 26 questions
5. Logarithmic functions and the natural logarithm 21 questions
6. Differentials dy and dx 26 questions
7. Global and local extrema: definitions 20 questions
8. Proofs of the derivatives of inverse trigonometric functions 21 questions
9. Chain rule: statement and Leibniz notation 25 questions
10. Stationary points 23 questions
11. Local extreme values 26 questions
12. Critical points 20 questions
13. Linearization: formula and meaning 23 questions
14. Increasing and decreasing functions: definitions 22 questions
15. Parametric derivative at a given value of t 24 questions
16. Implicit differentiation: method 24 questions
17. Equation of the tangent line 26 questions
18. Cylinder and open-box optimisation 28 questions
19. Exponential functions and the number e 21 questions
20. Second derivatives of functions 26 questions
21. Chain rule with powers and radicals 26 questions
22. Tangent and normal: further applications 24 questions
23. Approximations using differentials 26 questions
24. Intervals of increase and decrease: trigonometric and other functions 24 questions
25. Implicit differentiation at a point 23 questions
26. Differentiating inverse trigonometric expressions 26 questions
27. Estimating values by linear approximation 28 questions
28. Error, relative error and percentage error 24 questions
29. Implicit differentiation with trigonometric and exponential terms 24 questions
30. Differentiating exponential expressions 23 questions
31. Product rule 26 questions
32. Basic rules of differentiation 26 questions
33. Global extrema on a closed interval 24 questions
34. Differentiating logarithmic expressions 23 questions
35. nth derivatives of standard functions 26 questions
36. Formulas for derivatives of inverse trigonometric functions 21 questions
37. Steps for solving extrema problems 21 questions
38. Composition of functions 26 questions
39. Values of exponential derivatives 24 questions
40. Chain rule by substitution of u 26 questions
41. Horizontal and vertical tangents 20 questions
42. Actual change Δy compared with dy 26 questions
43. Window, wire, poster and triangle problems 22 questions
44. Computing local linear approximations 27 questions
45. Equation of the normal line 26 questions
46. Differentiating trigonometric expressions 26 questions
47. Verifying relations between derivatives 24 questions
48. Derivatives of sine and cosine by the definition 26 questions
49. Tangent line to a curve: slope 24 questions
50. Propagation of errors in measurements 28 questions
51. Sign of the derivative on sub-intervals 24 questions
52. Definition of the derivative 26 questions
53. Fencing and rectangle area problems 28 questions
54. Second derivative of parametric equations 24 questions
55. Simplifying with logarithm laws before differentiating 20 questions
56. Parametric differentiation 26 questions
57. Second derivative of implicit functions 24 questions
58. Values of logarithmic derivatives 24 questions
59. Intervals of increase and decrease of polynomials 20 questions
60. Rectangle of maximum area under a parabola 26 questions
61. Notation for higher-order derivatives 24 questions
62. Values of inverse trigonometric derivatives 26 questions
63. Third and fourth derivatives 21 questions
64. Minimum distance problems 24 questions
65. Derivatives of the six trigonometric functions 23 questions
66. Derivative of aˣ by the definition 24 questions
67. Derivative of logₐ x by the definition 22 questions

Sample questions: Further Differentiation

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Use implicit differentiation to find dy/dx if x² + y = exp(xy).

• A. (−2x + y exp(xy))/(−x exp(xy) + 2)
• B. (x exp(xy) − 1)/(2x − y exp(xy))
• C. (2x − y exp(xy))/(−x exp(xy) + 1)
• D. (−2x + y exp(xy))/(−x exp(xy) + 1)(correct answer)

Answer: D.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = 2x − y exp(xy) and F_y = −x exp(xy) + 1; this gives (−2x + y exp(xy))/(−x exp(xy) + 1).

Q2. Use implicit differentiation to find dy/dx if x² sin y = y² cos x.

• A. (2x sin y + y² sin x)/(x² cos y − 2y cos x)
• B. −2x sin y − y² sin x
• C. (−x² cos y + 2y cos x)/(2x sin y + y² sin x)
• D. (−2x sin y − y² sin x)/(x² cos y − 2y cos x)(correct answer)

Answer: D.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = 2x sin y + y² sin x and F_y = x² cos y − 2y cos x; this gives (−2x sin y − y² sin x)/(x² cos y − 2y cos x).

Q3. Use implicit differentiation to find dy/dx if x sin y + y cos x = 0.

• A. y sin x − sin y
• B. (−x cos y − cos x)/(−y sin x + sin y)
• C. (−y sin x + sin y)/(x cos y + cos x)
• D. (y sin x − sin y)/(x cos y + cos x)(correct answer)

Answer: D.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = −y sin x + sin y and F_y = x cos y + cos x; this gives (y sin x − sin y)/(x cos y + cos x).

Q4. Use implicit differentiation to find dy/dx if exp(xy) + tan y = x.

• A. (y exp(xy) − 1)/(x exp(xy) + sec² y)
• B. −y exp(xy) + 1
• C. (−x exp(xy) − sec² y)/(y exp(xy) − 1)
• D. (−y exp(xy) + 1)/(x exp(xy) + sec² y)(correct answer)

Answer: D.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = y exp(xy) − 1 and F_y = x exp(xy) + sec² y; this gives (−y exp(xy) + 1)/(x exp(xy) + sec² y).

Q5. Use implicit differentiation to find dy/dx if x + y² + sin(xy²) = 0.

• A. (y² cos(xy²) + 1)/(2xy cos(xy²) + 2y)
• B. (−2xy cos(xy²) − 2y)/(y² cos(xy²) + 1)
• C. (−y² cos(xy²) − 1)/(2xy cos(xy²) + 2y)(correct answer)
• D. −y² cos(xy²) − 1

Answer: C.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = y² cos(xy²) + 1 and F_y = 2xy cos(xy²) + 2y; this gives (−y² cos(xy²) − 1)/(2xy cos(xy²) + 2y).

Q6. Use implicit differentiation to find dy/dx if exp(x + y) = xy.

• A. (y − exp(x + y))/(−x + exp(x + y))(correct answer)
• B. (−y + exp(x + y))/(−x + exp(x + y))
• C. (x − exp(x + y))/(−y + exp(x + y))
• D. y − exp(x + y)

Answer: A.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = −y + exp(x + y) and F_y = −x + exp(x + y); this gives (y − exp(x + y))/(−x + exp(x + y)).

Q7. Use implicit differentiation to find dy/dx if y cos x = x sin y.

• A. (x cos y − cos x)/(−y sin x − sin y)
• B. (−y sin x − sin y)/(−x cos y + cos x)
• C. y sin x + sin y
• D. (y sin x + sin y)/(−x cos y + cos x)(correct answer)

Answer: D.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = −y sin x − sin y and F_y = −x cos y + cos x; this gives (y sin x + sin y)/(−x cos y + cos x).

Q8. Use implicit differentiation to find dy/dx if x cos y = y sin x.

• A. (y cos x − cos y)/(−x sin y − sin x)(correct answer)
• B. y cos x − cos y
• C. (−y cos x + cos y)/(−x sin y − sin x)
• D. (x sin y + sin x)/(−y cos x + cos y)

Answer: A.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = −y cos x + cos y and F_y = −x sin y − sin x; this gives (y cos x − cos y)/(−x sin y − sin x).

Q9. Use implicit differentiation to find dy/dx if sec(xy) = x.

• A. (y tan(xy) sec(xy) − 1)/(x tan(xy) sec(xy))
• B. (−y tan(xy) sec(xy) + 1)/(x tan(xy) sec(xy))(correct answer)
• C. −x tan(xy) sec(xy)/(y tan(xy) sec(xy) − 1)
• D. (−y tan(xy) sec(xy) + 1)/(x tan(xy) sec(xy) + 1)

Answer: B.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = y tan(xy) sec(xy) − 1 and F_y = x tan(xy) sec(xy); this gives (−y tan(xy) sec(xy) + 1)/(x tan(xy) sec(xy)).

Q10. Use implicit differentiation to find dy/dx if xy + cos(x + y) = 0.

• A. (y − sin(x + y))/(x − sin(x + y))
• B. (−y + sin(x + y))/(x − sin(x + y))(correct answer)
• C. −y + sin(x + y)
• D. (−x + sin(x + y))/(y − sin(x + y))

Answer: B.Differentiate both sides w.r.t. x and solve for dy/dx: dy/dx = −F_x/F_y, where F = LHS − RHS has F_x = y − sin(x + y) and F_y = x − sin(x + y); this gives (−y + sin(x + y))/(x − sin(x + y)).