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- 1. Sections 3.1–3.3 Derivatives of Exponential and Logarithmic Functions V63.0121.002.2010Su, Calculus I New York University June 1, 2010 Announcements Today: Homework 2 due Tomorrow: Section 3.4, review Thursday: Midterm in class . . . . . .
- 2. Announcements Today: Homework 2 due Tomorrow: Section 3.4, review Thursday: Midterm in class . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 2 / 54
- 3. Objectives for Sections 3.1 and 3.2 Know the definition of an exponential function Know the properties of exponential functions Understand and apply the laws of logarithms, including the change of base formula. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 3 / 54
- 4. Objectives for Section 3.3 Know the derivatives of the exponential functions (with any base) Know the derivatives of the logarithmic functions (with any base) Use the technique of logarithmic differentiation to find derivatives of functions involving roducts, quotients, and/or exponentials. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 4 / 54
- 5. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 5 / 54
- 6. Derivation of exponential functions Definition If a is a real number and n is a positive whole number, then an = a · a · · · · · a n factors . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 6 / 54
- 7. Derivation of exponential functions Definition If a is a real number and n is a positive whole number, then an = a · a · · · · · a n factors Examples 23 = 2 · 2 · 2 = 8 34 = 3 · 3 · 3 · 3 = 81 (−1)5 = (−1)(−1)(−1)(−1)(−1) = −1 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 6 / 54
- 8. Fact If a is a real number, then ax+y = ax ay ax ax−y = y a (ax )y = axy (ab)x = ax bx whenever all exponents are positive whole numbers. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 7 / 54
- 9. Fact If a is a real number, then ax+y = ax ay ax ax−y = y a (ax )y = axy (ab)x = ax bx whenever all exponents are positive whole numbers. Proof. Check for yourself: ax+y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = ax ay x + y factors x factors y factors . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 7 / 54
- 10. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 8 / 54
- 11. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example: ! an = an+0 = an a0 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 8 / 54
- 12. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example: ! an = an+0 = an a0 Definition If a ̸= 0, we define a0 = 1. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 8 / 54
- 13. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example: ! an = an+0 = an a0 Definition If a ̸= 0, we define a0 = 1. Notice 00 remains undefined (as a limit form, it’s indeterminate). . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 8 / 54
- 14. Conventions for negative exponents If n ≥ 0, we want an · a−n = an+(−n) = a0 = 1 ! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 9 / 54
- 15. Conventions for negative exponents If n ≥ 0, we want an · a−n = an+(−n) = a0 = 1 ! Definition 1 If n is a positive integer, we define a−n = . an . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 9 / 54
- 16. Conventions for negative exponents If n ≥ 0, we want an · a−n = an+(−n) = a0 = 1 ! Definition 1 If n is a positive integer, we define a−n = . an Fact 1 The convention that a−n = “works” for negative n as well. an am If m and n are any integers, then am−n = n . a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 9 / 54
- 17. Conventions for fractional exponents If q is a positive integer, we want ! (a1/q )q = a1 = a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 10 / 54
- 18. Conventions for fractional exponents If q is a positive integer, we want ! (a1/q )q = a1 = a Definition √ If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q is even. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 10 / 54
- 19. Conventions for fractional exponents If q is a positive integer, we want ! (a1/q )q = a1 = a Definition √ If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q is even. √q ( √ )p Notice that ap = q a . So we can unambiguously say ap/q = (ap )1/q = (a1/q )p . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 10 / 54
- 20. Conventions for irrational powers So ax is well-defined if x is rational. What about irrational powers? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 11 / 54
- 21. Conventions for irrational powers So ax is well-defined if x is rational. What about irrational powers? Definition Let a > 0. Then ax = lim ar r→x r rational . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 11 / 54
- 22. Conventions for irrational powers So ax is well-defined if x is rational. What about irrational powers? Definition Let a > 0. Then ax = lim ar r→x r rational In other words, to approximate ax for irrational x, take r close to x but rational and compute ar . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 11 / 54
- 23. Graphs of various exponential functions y . . x . . . . . . .
- 24. Graphs of various exponential functions y . . = 1x y . x . . . . . . .
- 25. Graphs of various exponential functions y . . = 2x y . = 1x y . x . . . . . . .
- 26. Graphs of various exponential functions y . . = 3x. = 2x y y . = 1x y . x . . . . . . .
- 27. Graphs of various exponential functions y . . = 10x= 3x. = 2x y y . y . = 1x y . x . . . . . . .
- 28. Graphs of various exponential functions y . . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . .
- 29. Graphs of various exponential functions y . . = (1/2)x y . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . .
- 30. Graphs of various exponential functions x y . . = (1/2)x (1/3) y y . = . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . .
- 31. Graphs of various exponential functions y . y . = x . = (1/2)x (1/3) y . = (1/10)x. = 10x= 3x. = 2x y y y . y . = 1.5x y . = 1x y . x . . . . . . .
- 32. Graphs of various exponential functions y . y yx .. = ((1/2)x (1/3)x y = 2/. )= 3 . = (1/10)x. = 10x= 3x. = 2x y y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 12 / 54
- 33. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 13 / 54
- 34. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain R and range (0, ∞). In particular, ax > 0 for all x. If a, b > 0 and x, y ∈ R, then ax+y = ax ay ax ax−y = y a (ax )y = axy (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 14 / 54
- 35. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain R and range (0, ∞). In particular, ax > 0 for all x. If a, b > 0 and x, y ∈ R, then ax+y = ax ay ax ax−y = y negative exponents mean reciprocals. a (ax )y = axy (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 14 / 54
- 36. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain R and range (0, ∞). In particular, ax > 0 for all x. If a, b > 0 and x, y ∈ R, then ax+y = ax ay ax ax−y = y negative exponents mean reciprocals. a (ax )y = axy fractional exponents mean roots (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 14 / 54
- 37. Simplifying exponential expressions Example Simplify: 82/3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 15 / 54
- 38. Simplifying exponential expressions Example Simplify: 82/3 Solution √ 3 √ 82/3 = 82 = 3 64 = 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 15 / 54
- 39. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 15 / 54
- 40. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, Example √ 8 Simplify: 21/2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 15 / 54
- 41. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, Example √ 8 Simplify: 21/2 Answer 2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 15 / 54
- 42. Limits of exponential functions Fact (Limits of exponential y . functions) . = (= 2()1/32/3)x y . 1/ =x( )x y . y y y = x . 3x y . = (. /10)10x= 2x. = 1 . = y y If a > 1, then lim ax = ∞ x→∞ and lim ax = 0 x→−∞ If 0 < a < 1, then lim ax = 0 and y . = x→∞ lim a = ∞ x . x . x→−∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 16 / 54
- 43. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 17 / 54
- 44. Compounded Interest Question Suppose you save $100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 18 / 54
- 45. Compounded Interest Question Suppose you save $100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer $100 + 10% = $110 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 18 / 54
- 46. Compounded Interest Question Suppose you save $100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer $100 + 10% = $110 $110 + 10% = $110 + $11 = $121 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 18 / 54
- 47. Compounded Interest Question Suppose you save $100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer $100 + 10% = $110 $110 + 10% = $110 + $11 = $121 $100(1.1)t . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 18 / 54
- 48. Compounded Interest: quarterly Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 19 / 54
- 49. Compounded Interest: quarterly Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer $100(1.025)4 = $110.38, . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 19 / 54
- 50. Compounded Interest: quarterly Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer $100(1.025)4 = $110.38, not $100(1.1)4 ! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 19 / 54
- 51. Compounded Interest: quarterly Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer $100(1.025)4 = $110.38, not $100(1.1)4 ! $100(1.025)8 = $121.84 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 19 / 54
- 52. Compounded Interest: quarterly Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer $100(1.025)4 = $110.38, not $100(1.1)4 ! $100(1.025)8 = $121.84 $100(1.025)4t . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 19 / 54
- 53. Compounded Interest: monthly Question Suppose you save $100 at 10% annual interest, with interest compounded twelve times a year. How much do you have after t years? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 20 / 54
- 54. Compounded Interest: monthly Question Suppose you save $100 at 10% annual interest, with interest compounded twelve times a year. How much do you have after t years? Answer $100(1 + 10%/12)12t . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 20 / 54
- 55. Compounded Interest: general Question Suppose you save P at interest rate r, with interest compounded n times a year. How much do you have after t years? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 21 / 54
- 56. Compounded Interest: general Question Suppose you save P at interest rate r, with interest compounded n times a year. How much do you have after t years? Answer ( r )nt B(t) = P 1 + n . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 21 / 54
- 57. Compounded Interest: continuous Question Suppose you save P at interest rate r, with interest compounded every instant. How much do you have after t years? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 22 / 54
- 58. Compounded Interest: continuous Question Suppose you save P at interest rate r, with interest compounded every instant. How much do you have after t years? Answer ( ( ) r )nt 1 rnt B(t) = lim P 1 + = lim P 1 + n→∞ n n→∞ n [ ( )n ]rt 1 =P lim 1 + n→∞ n independent of P, r, or t . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 22 / 54
- 59. The magic number Definition ( ) 1 n e = lim 1 + n→∞ n . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 23 / 54
- 60. The magic number Definition ( ) 1 n e = lim 1 + n→∞ n So now continuously-compounded interest can be expressed as B(t) = Pert . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 23 / 54
- 61. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 62. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 63. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 64. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 65. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 1000 2.71692 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 66. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 67. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 68. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 e is irrational 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 69. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 e is irrational 100 2.70481 1000 2.71692 e is transcendental 106 2.71828 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 24 / 54
- 70. Meet the Mathematician: Leonhard Euler Born in Switzerland, lived in Prussia (Germany) and Russia Eyesight trouble all his life, blind from 1766 onward Hundreds of contributions to calculus, number theory, graph theory, fluid mechanics, optics, and astronomy Leonhard Paul Euler Swiss, 1707–1783 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 25 / 54
- 71. A limit . Question eh − 1 What is lim ? h→0 h . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 26 / 54
- 72. A limit . Question eh − 1 What is lim ? h→0 h Answer If h is small enough, e ≈ (1 + h)1/h . So eh − 1 ≈1 h . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 26 / 54
- 73. A limit . Question eh − 1 What is lim ? h→0 h Answer If h is small enough, e ≈ (1 + h)1/h . So eh − 1 ≈1 h eh − 1 In fact, lim = 1. h→0 h 2h − 1 This can be used to characterize e: lim = 0.693 · · · < 1 and h→0 h 3h − 1 lim = 1.099 · · · > 1 h→0 h . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 26 / 54
- 74. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 27 / 54
- 75. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 28 / 54
- 76. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 28 / 54
- 77. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 28 / 54
- 78. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x (iii) loga (xr ) = r loga x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 28 / 54
- 79. Logarithms convert products to sums Suppose y = loga x and y′ = loga x′ ′ Then x = ay and x′ = ay ′ ′ So xx′ = ay ay = ay+y Therefore loga (xx′ ) = y + y′ = loga x + loga x′ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 29 / 54
- 80. Example Write as a single logarithm: 2 ln 4 − ln 3. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 30 / 54
- 81. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 30 / 54
- 82. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 30 / 54
- 83. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 Answer ln 12 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 30 / 54
- 84. “ . . lawn” . . . . . . . . Image credit: Selva V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 31 / 54
- 85. Graphs of logarithmic functions y . . = 2x y y . = log2 x . . 0, 1) ( ..1, 0) . ( x . . . . . . .
- 86. Graphs of logarithmic functions y . . = 3x= 2x y . y y . = log2 x y . = log3 x . . 0, 1) ( ..1, 0) . ( x . . . . . . .
- 87. Graphs of logarithmic functions y . . = .10x 3x= 2x y y= . y y . = log2 x y . = log3 x . . 0, 1) ( y . = log10 x ..1, 0) . ( x . . . . . . .
- 88. Graphs of logarithmic functions y . . = .10=3x= 2x y xy y y. = .ex y . = log2 x y . = ln x y . = log3 x . . 0, 1) ( y . = log10 x ..1, 0) . ( x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 32 / 54
- 89. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 33 / 54
- 90. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a Proof. If y = loga x, then x = ay So ln x = ln(ay ) = y ln a Therefore ln x y = loga x = ln a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 33 / 54
- 91. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 34 / 54
- 92. Derivatives of Exponential Functions Fact If f(x) = ax , then f′ (x) = f′ (0)ax . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 35 / 54
- 93. Derivatives of Exponential Functions Fact If f(x) = ax , then f′ (x) = f′ (0)ax . Proof. Follow your nose: f(x + h) − f(x) ax+h − ax f′ (x) = lim = lim h→0 h h→0 h a x ah − ax a h−1 = lim = ax · lim = ax · f′ (0). h→0 h h→0 h . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 35 / 54
- 94. Derivatives of Exponential Functions Fact If f(x) = ax , then f′ (x) = f′ (0)ax . Proof. Follow your nose: f(x + h) − f(x) ax+h − ax f′ (x) = lim = lim h→0 h h→0 h a x ah − ax a h−1 = lim = ax · lim = ax · f′ (0). h→0 h h→0 h To reiterate: the derivative of an exponential function is a constant times that function. Much different from polynomials! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 35 / 54
- 95. The funny limit in the case of e Remember the definition of e: ( ) 1 n e = lim 1 + = lim (1 + h)1/h n→∞ n h→0 Question eh − 1 What is lim ? h→0 h . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 36 / 54
- 96. The funny limit in the case of e Remember the definition of e: ( ) 1 n e = lim 1 + = lim (1 + h)1/h n→∞ n h→0 Question eh − 1 What is lim ? h→0 h Answer If h is small enough, e ≈ (1 + h)1/h . So [ ]h eh − 1 (1 + h)1/h − 1 (1 + h) − 1 h ≈ = = =1 h h h h . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 36 / 54
- 97. The funny limit in the case of e Remember the definition of e: ( ) 1 n e = lim 1 + = lim (1 + h)1/h n→∞ n h→0 Question eh − 1 What is lim ? h→0 h Answer If h is small enough, e ≈ (1 + h)1/h . So [ ]h eh − 1 (1 + h)1/h − 1 (1 + h) − 1 h ≈ = = =1 h h h h eh − 1 So in the limit we get equality: lim =1 h→0 h . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 36 / 54
- 98. Derivative of the natural exponential function From ( ) d x ah − 1 eh − 1 a = lim ax and lim =1 dx h→0 h h→0 h we get: Theorem d x e = ex dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 37 / 54
- 99. Exponential Growth Commonly misused term to say something grows exponentially It means the rate of change (derivative) is proportional to the current value Examples: Natural population growth, compounded interest, social networks . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 38 / 54
- 100. Examples Examples Find these derivatives: e3x 2 ex x2 ex . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 39 / 54
- 101. Examples Examples Find these derivatives: e3x 2 ex x2 ex Solution d 3x e = 3e3x dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 39 / 54
- 102. Examples Examples Find these derivatives: e3x 2 ex x2 ex Solution d 3x e = 3e3x dx d x2 2 d 2 e = ex (x2 ) = 2xex dx dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 39 / 54
- 103. Examples Examples Find these derivatives: e3x 2 ex x2 ex Solution d 3x e = 3e3x dx d x2 2 d 2 e = ex (x2 ) = 2xex dx dx d 2 x x e = 2xex + x2 ex dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 39 / 54
- 104. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 40 / 54
- 105. Derivative of the natural logarithm function Let y = ln x. Then x = ey so . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 106. Derivative of the natural logarithm function Let y = ln x. Then x = ey so dy ey =1 dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 107. Derivative of the natural logarithm function Let y = ln x. Then x = ey so dy ey=1 dx dy 1 1 =⇒ = y = dx e x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 108. Derivative of the natural logarithm function Let y = ln x. Then x = ey so dy ey=1 dx dy 1 1 =⇒ = y = dx e x So: Fact d 1 ln x = dx x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 109. Derivative of the natural logarithm function y . Let y = ln x. Then x = ey so dy ey=1 dx l .n x dy 1 1 =⇒ = y = dx e x . x . So: Fact d 1 ln x = dx x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 110. Derivative of the natural logarithm function y . Let y = ln x. Then x = ey so dy ey=1 dx l .n x dy 1 1 1 =⇒ = y = . dx e x x . x . So: Fact d 1 ln x = dx x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 41 / 54
- 111. The Tower of Powers y y′ x3 3x2 The derivative of a power 2 1 function is a power function x 2x of one lower power x1 1x0 x0 0 ? ? x−1 −1x−2 x−2 −2x−3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 42 / 54
- 112. The Tower of Powers y y′ x3 3x2 The derivative of a power 2 1 function is a power function x 2x of one lower power x1 1x0 Each power function is the x 0 0 derivative of another power function, except x−1 ? x−1 x−1 −1x−2 x−2 −2x−3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 42 / 54
- 113. The Tower of Powers y y′ x3 3x2 The derivative of a power 2 1 function is a power function x 2x of one lower power x1 1x0 Each power function is the x 0 0 derivative of another power function, except x−1 ln x x−1 ln x fills in this gap x−1 −1x−2 precisely. x−2 −2x−3 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 42 / 54
- 114. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 43 / 54
- 115. Other logarithms Example d x Use implicit differentiation to find a . dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 44 / 54
- 116. Other logarithms Example d x Use implicit differentiation to find a . dx Solution Let y = ax , so ln y = ln ax = x ln a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 44 / 54
- 117. Other logarithms Example d x Use implicit differentiation to find a . dx Solution Let y = ax , so ln y = ln ax = x ln a Differentiate implicitly: 1 dy dy = ln a =⇒ = (ln a)y = (ln a)ax y dx dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 44 / 54
- 118. Other logarithms Example d x Use implicit differentiation to find a . dx Solution Let y = ax , so ln y = ln ax = x ln a Differentiate implicitly: 1 dy dy = ln a =⇒ = (ln a)y = (ln a)ax y dx dx Before we showed y′ = y′ (0)y, so now we know that 2h − 1 3h − 1 ln 2 = lim ≈ 0.693 ln 3 = lim ≈ 1.10 h→0 h h→0 h . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 44 / 54
- 119. Other logarithms Example d Find loga x. dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 45 / 54
- 120. Other logarithms Example d Find loga x. dx Solution Let y = loga x, so ay = x. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 45 / 54
- 121. Other logarithms Example d Find loga x. dx Solution Let y = loga x, so ay = x. Now differentiate implicitly: dy dy 1 1 (ln a)ay = 1 =⇒ = y = dx dx a ln a x ln a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 45 / 54
- 122. Other logarithms Example d Find loga x. dx Solution Let y = loga x, so ay = x. Now differentiate implicitly: dy dy 1 1 (ln a)ay = 1 =⇒ = y = dx dx a ln a x ln a Another way to see this is to take the natural logarithm: ln x ay = x =⇒ y ln a = ln x =⇒ y = ln a dy 1 1 So = . dx ln a x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 45 / 54
- 123. More examples Example d Find log2 (x2 + 1) dx . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 46 / 54
- 124. More examples Example d Find log2 (x2 + 1) dx Answer dy 1 1 2x = 2+1 (2x) = dx ln 2 x (ln 2)(x2 + 1) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 46 / 54
- 125. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions Derivatives of Exponential Functions Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Differentiation The power rule for irrational powers . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 47 / 54
- 126. A nasty derivative Example √ (x2 + 1) x + 3 Let y = . Find y′ . x−1 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 48 / 54
- 127. A nasty derivative Example √ (x2 + 1) x + 3 Let y = . Find y′ . x−1 Solution We use the quotient rule, and the product rule in the numerator: [ √ ] √ (x − 1) 2x x + 3 + (x2 + 1) 1 (x + 3)−1/2 − (x2 + 1) x + 3(1) 2 y′ = (x − 1)2 √ √ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 48 / 54
- 128. Another way √ (x2 + 1) x + 3 y= x−1 1 ln y = ln(x2 + 1) + ln(x + 3) − ln(x − 1) 2 1 dy 2x 1 1 = 2 + − y dx x + 1 2(x + 3) x − 1 So ( ) dy 2x 1 1 = + − y dx x2+1 2(x + 3) x − 1 ( ) √ 2x 1 1 (x2 + 1) x + 3 = + − x2 + 1 2(x + 3) x − 1 x−1 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 49 / 54
- 129. Compare and contrast Using the product, quotient, and power rules: √ √ ′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 y = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2 Using logarithmic differentiation: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 x−1 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 50 / 54
- 130. Compare and contrast Using the product, quotient, and power rules: √ √ ′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 y = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2 Using logarithmic differentiation: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 x−1 Are these the same? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 50 / 54
- 131. Compare and contrast Using the product, quotient, and power rules: √ √ ′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 y = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2 Using logarithmic differentiation: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 x−1 Are these the same? Which do you like better? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 50 / 54
- 132. Compare and contrast Using the product, quotient, and power rules: √ √ ′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 y = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2 Using logarithmic differentiation: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 x−1 Are these the same? Which do you like better? What kinds of expressions are well-suited for logarithmic differentiation? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 50 / 54
- 133. Derivatives of powers Let y = xx . Which of these is true? (A) Since y is a power function, y′ = x · xx−1 = xx . (B) Since y is an exponential function, y′ = (ln x) · xx (C) Neither . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 51 / 54
- 134. Derivatives of powers Let y = xx . Which of these is true? (A) Since y is a power function, y′ = x · xx−1 = xx . (B) Since y is an exponential function, y′ = (ln x) · xx (C) Neither . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 51 / 54
- 135. It's neither! Or both? If y = xx , then ln y = x ln x 1 dy 1 = x · + ln x = 1 + ln x y dx x dy = xx + (ln x)xx dx Each of these terms is one of the wrong answers! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 52 / 54
- 136. Derivative of arbitrary powers Fact (The power rule) Let y = xr . Then y′ = rxr−1 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 53 / 54
- 137. Derivative of arbitrary powers Fact (The power rule) Let y = xr . Then y′ = rxr−1 . Proof. y = xr =⇒ ln y = r ln x Now differentiate: 1 dy r = y dx x dy y =⇒ = r = rxr−1 dx x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 53 / 54
- 138. Summary . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Exponential and Logarithmic June 1, 2010 54 / 54

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