Basic Calculus Basic Differentiation Rules
- 1. Capstone Project Science, Technology, Engineering, and Mathematics Basic Calculus Science, Technology, Engineering, and Mathematics Lesson 5.1 The Basic Differentiation Rules
- 2. 2 A car speedometer tells us our current speed when we drive. However, this speed is not either our constant speed nor our average speed when we drive.
- 3. 3 This speed is called the instantaneous rate of change, or simply the instantaneous speed. This concept is related to limits and derivatives.
- 4. 4 In this lesson, we are going to derive basic differentiation rules with the aid of the limit definition of derivative to enable us to solve for the derivatives of algebraic functions.
- 5. 5 How do we differentiate functions without the use of the limit definition of derivative?
- 6. Learning Competencies At the end of the lesson, you should be able to do the following: 6 ● Derive the differentiation rules (STEM_BC11D- IIIf-2). ● Apply the differentiation rules in computing the derivative of an algebraic, exponential, and trigonometric functions (STEM_BC11D-IIIf- 3).
- 7. Learning Objectives At the end of the lesson, you should be able to do the following: 7 ● Derive the basic differentiation rules (Constant Rule, Power Rule, Constant Multiple Rule, Sum or Difference Rule). ● Apply the basic differentiation rules in solving for the derivatives of functions.
- 8. 8 Constant Rule The derivative of a constant function is 0. If 𝑐 is a constant, then 𝒅 𝒅𝒙 𝒄 = 𝟎. The Basic Differentiation Rules
- 9. 9 Proof: Let 𝑓 𝑥 = 𝑐. We use the limit definition of a derivative to derive the Constant Rule. The Basic Differentiation Rules
- 10. 10 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ Limit Definition of Derivative
- 11. 11 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ = lim ℎ→0 𝑐 − 𝑐 ℎ = lim ℎ→0 0 Limit Definition of Derivative Subtraction
- 12. 12 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ = lim ℎ→0 𝑐 − 𝑐 ℎ = lim ℎ→0 0 = 0 Limit Definition of Derivative Subtraction Constant Law of Limits
- 13. 13 Proof: Therefore, 𝑑 𝑑𝑥 𝑐 = 0. The Basic Differentiation Rules
- 14. 14 Example: a. If 𝑓 𝑥 = 6, then 𝑓′ 𝑥 = 0. b. Given than 𝑦 = 24, 𝑦′ = 0. The Basic Differentiation Rules
- 15. 15 Power Rule If 𝑘 is a real number, then the derivative of the function 𝑓 𝑥 = 𝑥𝑘 is 𝒅 𝒅𝒙 𝒙𝒌 = 𝒌𝒙𝒌−𝟏 . The Basic Differentiation Rules
- 16. 16 Proof: We will prove the Power Rule for positive integer exponent. The proof for negative integer and real number exponents need more advanced techniques (Quotient Rule and implicit differentiation, respectively) which will be discussed in the succeeding lessons on derivatives. The Basic Differentiation Rules
- 17. 17 Proof: Let 𝑓 𝑥 = 𝑥𝑘, where 𝑘 is any positive integer. We use the limit definition of a derivative to derive the Power Rule. The Basic Differentiation Rules
- 18. 18 The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑥𝑘 = lim ℎ→0 𝑥 + ℎ 𝑘 − 𝑥𝑘 ℎ Limit Definition of Derivative
- 19. 19 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑥𝑘 = lim ℎ→0 𝑥𝑘 + 𝑘𝑥𝑘−1 ℎ + 𝑘 𝑘 − 1 2 𝑥𝑘−2 ℎ2 + ⋯ + 𝑘𝑥ℎ𝑘−1 + ℎ𝑘 − 𝑥𝑘 ℎ Binomial Expansion
- 20. 20 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑥𝑘 = lim ℎ→0 𝑘𝑥𝑘−1ℎ + 𝑘 𝑘 − 1 2 𝑥𝑘−2ℎ2 + ⋯ + 𝑘𝑥ℎ𝑘−1 + ℎ𝑘 ℎ Subtraction
- 21. 21 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑥𝑘 = lim ℎ→0 ℎ 𝑘𝑥𝑘−1 + 𝑘 𝑘 − 1 2 𝑥𝑘−2ℎ + ⋯ + 𝑘𝑥ℎ𝑘−2 + ℎ𝑘−1 ℎ Factor and cancel ℎ.
- 22. 22 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑥𝑘 = lim ℎ→0 𝑘𝑥𝑘−1 + 𝑘 𝑘 − 1 𝑥𝑘−2 2 𝑥𝑘−2 ℎ + ⋯ + 𝑘𝑥ℎ𝑘−2 + ℎ𝑘−1 = 𝑘𝑥𝑘−1 Evaluation of limit
- 23. 23 Proof: Therefore, 𝑑 𝑑𝑥 𝑥𝑘 = 𝑘𝑥𝑘−1. The Basic Differentiation Rules
- 24. 24 Example: Find the derivative of 𝑓 𝑥 = 𝑥4. The Basic Differentiation Rules
- 25. 25 Example: Find the derivative of 𝑓 𝑥 = 𝑥4. The derivative of 𝑓 𝑥 = 𝑥4 is 𝑓′ 𝑥 = 4𝑥3 . The Basic Differentiation Rules
- 26. 26 Constant Multiple Rule If 𝑓 is differentiable and 𝑐 is a constant, then the derivative of the function 𝑐 ∙ 𝑓 is 𝒅 𝒅𝒙 𝒄𝒇 𝒙 = 𝒄𝒇′ 𝒙 . The Basic Differentiation Rules
- 27. 27 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = lim ℎ→0 𝑐𝑓 𝑥 + ℎ − 𝑐𝑓 𝑥 ℎ Limit Definition of Derivative
- 28. 28 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = lim ℎ→0 𝑐 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ Factor 𝑐.
- 29. 29 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = 𝑐 lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ Constant Multiple Law for Limits
- 30. 30 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = 𝑐𝑓′ 𝑥 Substitution: 𝑓′ 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ
- 31. 31 Proof: Therefore, 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = 𝑐𝑓′ 𝑥 . The Basic Differentiation Rules
- 32. 32 Example: Use the Constant Multiple Rule to find the derivative of 𝑓 𝑥 = 4𝑥6. The Basic Differentiation Rules
- 33. 33 Example: Use the Constant Multiple Rule to find the derivative of 𝑓 𝑥 = 4𝑥6. The derivative of 𝑓 𝑥 = 4𝑥6 is 𝑓′ 𝑥 = 24𝑥5. The Basic Differentiation Rules
- 34. 34 Sum or Difference Rule Let 𝑓 and 𝑔 be differentiable functions, then the sum (or difference) of 𝑓 and 𝑔 is also differentiable. The derivative of the sum of 𝑓 and 𝑔 is given by 𝒅 𝒅𝒙 𝒇 𝒙 + 𝒈 𝒙 = 𝒇′ 𝒙 + 𝒈′ 𝒙 . The Basic Differentiation Rules
- 35. 35 Sum or Difference Rule The derivative of the sum of 𝑓 and 𝑔 is given by 𝒅 𝒅𝒙 𝒇 𝒙 − 𝒈 𝒙 = 𝒇′ 𝒙 − 𝒈′ 𝒙 . The Basic Differentiation Rules
- 36. 36 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ + 𝑔 𝑥 + ℎ − 𝑓 𝑥 + 𝑔 𝑥 ℎ Limit Definition of Derivative
- 37. 37 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ + 𝑔 𝑥 + ℎ − 𝑓 𝑥 − 𝑔 𝑥 ℎ Distributivity
- 38. 38 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 + 𝑔 𝑥 + ℎ − 𝑔 𝑥 ℎ Associativity
- 39. 39 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = lim ℎ→0 𝑓 𝑥 + ℎ − 𝑓 𝑥 ℎ + lim ℎ→0 𝑔 𝑥 + ℎ − 𝑔 𝑥 ℎ Sum or Difference Law for Limits
- 40. 40 Proof: The Basic Differentiation Rules 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = 𝑓′ 𝑥 + 𝑔′(𝑥) Limit Definition of Derivative
- 41. 41 Example: Use the Sum or Difference Rule to find the derivative of 𝑓 𝑥 = 𝑥3 + 𝑥2. The Basic Differentiation Rules
- 42. 42 Example: Use the Sum or Difference Rule to find the derivative of 𝑓 𝑥 = 𝑥3 + 𝑥2. The derivative of the function 𝑓 𝑥 = 𝑥3 + 𝑥2 is 𝑓′ 𝑥 = 3𝑥2 + 2𝑥. The Basic Differentiation Rules
- 43. 43 What differentiation rules can be used to get the derivative of 𝒇 𝒙 = 𝒙𝟑 + 𝟑𝒙𝟐 + 𝟓?
- 44. Let’s Practice! 44 Find the derivative of 𝒚 = 𝟒𝒙𝟕
- 45. Let’s Practice! 45 Find the derivative of 𝒚 = 𝟒𝒙𝟕 𝒚′ = 𝟐𝟖𝒙𝟔
- 46. Try It! 46 46 Find the derivative of 𝒚 = −𝟓𝒙𝟗 .
- 47. Tips 47 In differentiating functions of the form 𝒇 𝒙 = 𝒄 ∙ 𝒙𝒌 , we can find the derivative of the function by multiplying the exponent to the coefficient of the function and subtracting 1 from the exponent.
- 48. Tips 48 To differentiate the function 𝒇 𝒙 = 𝟑𝒙𝟔, we can directly multiply the exponent 6 to the coefficient 3 and subtract 1 from the exponent. Thus, the derivative of the function will be 𝒇′ 𝒙 = 𝟏𝟖𝒙𝟓 .
- 49. Let’s Practice! 49 What is the derivative of the function 𝒇 𝒙 = 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟖
- 50. Let’s Practice! 50 What is the derivative of the function 𝒇 𝒙 = 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟖 𝒇′ 𝒙 = 𝟑𝒙𝟐 + 𝟔𝒙 − 𝟔
- 51. Try It! 51 51 What is the derivative of the function 𝒇 𝒙 = 𝟐𝒙𝟑 − 𝟗𝒙𝟐 + 𝟒𝒙 − 𝟔?
- 52. Let’s Practice! 52 Find the equation of the line tangent to the curve 𝒚 = 𝒙𝟑 − 𝒙𝟐 at 𝒙 = 𝟏.
- 53. Let’s Practice! 53 Find the equation of the line tangent to the curve 𝒚 = 𝒙𝟑 − 𝒙𝟐 at 𝒙 = 𝟏. 𝒚 = 𝒙 − 𝟏 or 𝒙 − 𝒚 = 𝟏
- 54. Try It! 54 54 Find the equation of the line tangent to the curve 𝒚 = 𝒙𝟒 − 𝒙𝟐 at 𝒙 = −𝟏.
- 55. Let’s Practice! 55 Find the instantaneous rate of change in 𝒇 𝒙 = 𝟑𝒙𝟑 + 𝒙𝟐 − 𝟓 with respect to 𝒙 at the instant when 𝒙 = 𝟏.
- 56. Let’s Practice! 56 Find the instantaneous rate of change in 𝒇 𝒙 = 𝟑𝒙𝟑 + 𝒙𝟐 − 𝟓 with respect to 𝒙 at the instant when 𝒙 = 𝟏. 𝟏𝟏
- 57. Try It! 57 57 Find the instantaneous rate of change in 𝒇 𝒙 = 𝟐𝒙𝟑 − 𝟒𝒙𝟐 − 𝟗 with respect to 𝒙 at the instant when 𝒙 = −𝟐.
- 58. Remember 58 The first derivative of the function 𝒇 𝒙 tells us about the instantaneous rate of change when 𝒙 = 𝒄, where 𝒄 is a constant. ● If 𝒇′ 𝒙 > 𝟎, the function is increasing at the instant when 𝒙 = 𝒄. ● If 𝒇′ 𝒙 < 𝟎, the function is decreasing at the instant when 𝒙 = 𝒄.
- 59. Check Your Understanding 59 Find the derivative of each function. 1. 𝑓 𝑥 = 26 2. 𝑦 = 4 − 2𝑥 3. 𝑔 𝑥 = −9𝑥18 4. 𝑓 𝑥 = 1 𝑥3 5. 𝑦 = 𝑥2 + 𝑥 + 1 𝑥 − 1
- 60. Check Your Understanding 60 Solve the following problems. 1. What is the instantaneous rate of change for the function 𝑦 = 3𝑥2 − 𝑥 − 2 when 𝑥 = 2? 2. On which points on the curve 𝑦 = 𝑥3 − 3𝑥2 + 4 are the tangent lines horizontal?
- 61. Let’s Sum It Up! 61 The following are the Basic Differentiation Rules, which were all derived from the limit definition of derivatives. ● Constant Rule 𝑑 𝑑𝑥 𝑐 = 0
- 62. Let’s Sum It Up! 62 ● Power Rule 𝑑 𝑑𝑥 𝑥𝑘 = 𝑘𝑥𝑘−1
- 63. Let’s Sum It Up! 63 ● Constant Multiple Rule 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = 𝑐 ∙ 𝑓′ 𝑥
- 64. Let’s Sum It Up! 64 ● Sum or Difference Rule 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = 𝑓′ 𝑥 + 𝑔′ 𝑥 𝑑 𝑑𝑥 𝑓 𝑥 − 𝑔 𝑥 = 𝑓′ 𝑥 − 𝑔′ 𝑥
- 65. Key Formulas 65 Concept Formula Description Constant Rule 𝑑 𝑑𝑥 𝑐 = 0 Use this formula to solve for the derivative of a constant function. Power Rule 𝑑 𝑑𝑥 𝑥𝑘 = 𝑘𝑥𝑘−1 Use this formula to solve for the derivative of a variable raised to a real number exponent.
- 66. Key Formulas 66 Concept Formula Description Constant Multiple Rule 𝑑 𝑑𝑥 𝑐𝑓 𝑥 = 𝑐𝑓′ 𝑥 Use this formula to solve for the derivative of a constant times a function. Sum or Difference Rule 𝑑 𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥 = 𝑓′ 𝑥 + 𝑔′ 𝑥 𝑑 𝑑𝑥 𝑓 𝑥 − 𝑔 𝑥 = 𝑓′ 𝑥 − 𝑔′ 𝑥 Use this formula to solve for the derivative of sum or difference of functions.
- 67. Challenge Yourself 67 67 In the function 𝒇 𝒙 = 𝒂𝒙𝟐 − 𝒙 − 𝟏 , what value of 𝒂 will make 𝒇′ 𝟐 = 𝟕?
- 68. Bibliography 68 Edwards, C.H., and David E. Penney. Calculus: Early Transcendentals. 7th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008. Larson, Ron H., and Bruce H. Edwards. Essential Calculus: Early Transcendental Functions. Boston: Houghton Mifflin, 2008. Leithold, Louis. The Calculus 7. New York: HarperCollins College Publ., 1997. Smith, Robert T., and Roland B. Milton. Calculus. New York: McGraw Hill, 2012. Tan, Soo T. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach. Australia: Brooks/Cole Cengage Learning, 2012.