Sec on 3.3    Deriva ves of Logarithmic and        Exponen al Func ons            V63.0121.011: Calculus I          Profes...
Announcements   Quiz 3 next week on 2.6,   2.8, 3.1, 3.2
Objectives   Know the deriva ves of the   exponen al func ons (with any   base)   Know the deriva ves of the   logarithmic...
Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on    Exponen al Growth Deriva ve of the natural lo...
Conventions on power expressions Let a be a posi ve real number.      If n is a posi ve whole number, then an = a · a · · ...
Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and rang...
Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and rang...
Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and rang...
Graphs of exponential functions                          y      y y =y/=3(1/3)x        = (1(2/x)x             2)         y...
The magic number Defini on                      (          )n                             1            e = lim       1+    ...
Existence of eSee Appendix B                                      (      )n                                           1   ...
Logarithms Defini on    The base a logarithm loga x is the inverse of the func on ax                         y = loga x ⇐⇒ ...
Facts about Logarithms Facts   (i) loga (x1 · x2 ) = loga x1 + loga x2            ( )               x1  (ii) loga         ...
Graphs of logarithmic functions       y                   y =x ex               y =y10y3= 2x                     = x      ...
Change of base formula Fact If a > 0 and a ̸= 1, and the same for b, then                                       logb x    ...
Upshot of changing base The point of the change of base formula                    logb x   1         loga x =          = ...
Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on    Exponen al Growth Deriva ve of the natural lo...
Derivatives of Exponentials Fact If f(x) = ax , then f′ (x) = f′ (0)ax .
Derivatives of Exponentials Fact If f(x) = ax , then f′ (x) = f′ (0)ax . Proof. Follow your nose:            ′          f(...
The funny limit in the case of e Ques on            eh − 1 What is lim       ?        h→0    h Solu on
The funny limit in the case of e Ques on            eh − 1 What is lim       ?        h→0    h Solu on        (     )n    ...
The funny limit in the case of e Ques on             eh − 1 What is lim        ?         h→0    h Solu on So in the limit ...
Derivative of the naturalexponential function From                    (          )           d x          ah − 1          ...
Exponential Growth   Commonly misused term to say something grows exponen ally   It means the rate of change (deriva ve) i...
Examples Example     d Find e3x .     dx
Examples Example     d Find e3x .     dx Solu on              d 3x      d                 e = e3x (3x) = 3e3x             ...
Examples Example     d 2 Find ex .     dx
Examples Example     d 2 Find ex .     dx Solu on             d x2     2 d           2                e = ex (x2 ) = 2xex ...
Examples Example     d Find x2 ex .     dx
Examples Example     d Find x2 ex .     dx Solu on                d 2 x                   x e = 2xex + x2 ex              ...
Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on    Exponen al Growth Deriva ve of the natural lo...
Derivative ofy the natural logarithm Let y = ln x. Then x = e so
Derivative ofy the natural logarithm Let y = ln x. Then x = e so                 dy            ey      =1                 dx
Derivative ofy the natural logarithm Let y = ln x. Then x = e so            dy            ey =1            dx            d...
Derivative ofy the natural logarithm Let y = ln x. Then x = e so            dy            ey =1            dx            d...
Derivative ofy the natural logarithm Let y = ln x. Then x = e so                               y            dy            ...
Derivative ofy the natural logarithm Let y = ln x. Then x = e so                               y            dy            ...
The Tower of Powers   y    y′   x3   3x2   x2   2x1   The deriva ve of a power func on is a              power func on of ...
The Tower of Powers   y    y′   x3   3x2   x2   2x1   The deriva ve of a power func on is a              power func on of ...
The Tower of Powers   y     y′   x3    3x2   x2    2x1   The deriva ve of a power func on is a               power func on...
Examples Examples Find deriva ves of these func ons:      ln(3x)      x ln x         √      ln x
Examples Example      d Find    ln(3x).      dx
Examples Example      d Find    ln(3x).      dx Solu on (chain rule way)                      d           1      1        ...
Examples Example      d Find    ln(3x).      dx Solu on (proper es of logarithms way)             d           d           ...
Examples Example     d Find x ln x     dx
Examples Example     d Find x ln x     dx Solu on The product rule is in play here:               (     )         (       ...
Examples Example      d √ Find    ln x.      dx
Examples Example      d √ Find    ln x.      dx Solu on (chain rule way)                 d √       1 d√     1 1    1      ...
Examples Example      d √ Find    ln x.      dx Solu on (proper es of logarithms way)                            (        ...
Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on    Exponen al Growth Deriva ve of the natural lo...
Other logarithms Example                                    d x Use implicit differen a on to find      a.                  ...
Other logarithms Example                                    d x Use implicit differen a on to find      a.                  ...
Other logarithms Example                                       d x Use implicit differen a on to find         a.            ...
The funny limit in the case of a      x          ′  ′ Let y = e . Before we showed y = y (0)y, and now we know y′ = (ln a)...
Other logarithms Example      d Find   log x.      dx a
Other logarithms Example      d Find   log x.      dx a Solu on Let y = loga x, so ay = x.
Other logarithms Example      d Find   log x.      dx a Solu on Let y = loga x, so ay = x. Now differen ate implicitly:    ...
Other logarithms Example      d Find   log x.      dx a Solu on Or we can use the change of base formula:                 ...
More examples Example      d Find    log2 (x2 + 1)      dx
More examples Example      d Find    log2 (x2 + 1)      dx Answer                dy   1     1                2x           ...
Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on    Exponen al Growth Deriva ve of the natural lo...
A nasty derivative Example                 √         (x2 + 1) x + 3 Let y =                . Find y′ .              x−1
A nasty derivative Example                 √         (x2 + 1) x + 3 Let y =                . Find y′ .              x−1 So...
Another way                  √          (x2 + 1) x + 3       y=               x−1
Another way                     √             (x2 + 1) x + 3         y=                   x−1                         1   ...
Another way                      √              (x2 + 1) x + 3          y=                    x−1                         ...
Another way                         √                 (x2 + 1) x + 3             y=                       x−1             ...
Another way                         √                 (x2 + 1) x + 3             y=                       x−1             ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Compare and contrast   Using the product, quo ent, and power rules:               √                                    √  ...
Derivatives of powers Ques on                                y Let y = xx . Which of these is true? (A) Since y is a power...
Derivatives of powers Ques on                                y Let y = xx . Which of these is true? (A) Since y is a power...
Why not? Answer                                y (A) y′ ̸= xx because xx > 0 for all     x > 0, and this func on     decre...
Why not? Answer                                    y (A) y′ ̸= xx because xx > 0 for all     x > 0, and this func on     d...
It’s neither! Solu on If y = xx , then
It’s neither! Solu on If y = xx , then                    ln y = x ln x
It’s neither! Solu on If y = xx , then                     ln y = x ln x                    1 dy        1                 ...
It’s neither! Solu on If y = xx , then                     ln y = x ln x                    1 dy        1                 ...
Or both? Solu on  d x                                   y     x = xx + (ln x)xx = (1 + ln x)xx  dx                        ...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Or both? Solu on  d x                                    y     x = xx + (ln x)xx = (1 + ln x)xx  dx Remarks     Each of th...
Derivatives of power functionswith any exponent Fact (The power rule) Let y = xr . Then y′ = rxr−1 .
Derivatives of power functionswith any exponent Fact (The power rule) Let y = xr . Then y′ = rxr−1 . Proof.               ...
Summary   Deriva ves of               y          y′   Logarithmic and   Exponen al Func ons        ex          ex   Logari...
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Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)

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The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.

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Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)

  1. 1. Sec on 3.3 Deriva ves of Logarithmic and Exponen al Func ons V63.0121.011: Calculus I Professor Ma hew Leingang New York University March 21, 2011.
  2. 2. Announcements Quiz 3 next week on 2.6, 2.8, 3.1, 3.2
  3. 3. Objectives Know the deriva ves of the exponen al func ons (with any base) Know the deriva ves of the logarithmic func ons (with any base) Use the technique of logarithmic differen a on to find deriva ves of func ons involving roducts, quo ents, and/or exponen als.
  4. 4. Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on Exponen al Growth Deriva ve of the natural logarithm func on Deriva ves of other exponen als and logarithms Other exponen als Other logarithms Logarithmic Differen a on The power rule for irra onal powers
  5. 5. Conventions on power expressions Let a be a posi ve real number. If n is a posi ve whole number, then an = a · a · · · · · a n factors 0 a = 1. 1 For any real number r, a−r = . ar √ For any posi ve whole number n, a1/n = n a. There is only one con nuous func on which sa sfies all of the above. We call it the exponen al func on with base a.
  6. 6. Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x. For any real numbers x and y, and posi ve numbers a and b we have ax+y = ax ay x−y ax a = y a (a ) = axy x y (ab)x = ax bx
  7. 7. Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x. For any real numbers x and y, and posi ve numbers a and b we have ax+y = ax ay x−y ax a = y (nega ve exponents mean reciprocals) a (a ) = axy x y (ab)x = ax bx
  8. 8. Properties of exponentials Theorem If a > 0 and a ̸= 1, then f(x) = ax is a con nuous func on with domain (−∞, ∞) and range (0, ∞). In par cular, ax > 0 for all x. For any real numbers x and y, and posi ve numbers a and b we have ax+y = ax ay x−y ax a = y (nega ve exponents mean reciprocals) a (a ) = axy (frac onal exponents mean roots) x y (ab)x = ax bx
  9. 9. Graphs of exponential functions y y y =y/=3(1/3)x = (1(2/x)x 2) y = (1/10y x= 10x 3x = 2x ) y= y y = 1.5x y = 1x . x
  10. 10. The magic number Defini on ( )n 1 e = lim 1+ = lim+ (1 + h)1/h n→∞ n h→0
  11. 11. Existence of eSee Appendix B ( )n 1 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 irra onal 100 2.70481 1000 2.71692 e is transcendental 106 2.71828
  12. 12. Logarithms Defini on The base a logarithm loga x is the inverse of the func on ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey .
  13. 13. Facts about Logarithms Facts (i) loga (x1 · x2 ) = loga x1 + loga x2 ( ) x1 (ii) loga = loga x1 − loga x2 x2 (iii) loga (xr ) = r loga x
  14. 14. Graphs of logarithmic functions y y =x ex y =y10y3= 2x = x y = log2 x yy= log3 x = ln x (0, 1) y = log10 x . (1, 0) x
  15. 15. Change of base formula Fact If a > 0 and a ̸= 1, and the same for b, then logb x loga x = logb a
  16. 16. Upshot of changing base The point of the change of base formula logb x 1 loga x = = · logb x = (constant) · logb x logb a logb a is that all the logarithmic func ons are mul ples of each other. So just pick one and call it your favorite. Engineers like the common logarithm log = log10 Computer scien sts like the binary logarithm lg = log2 Mathema cians like natural logarithm ln = loge Naturally, we will follow the mathema cians. Just don’t pronounce it “lawn.”
  17. 17. Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on Exponen al Growth Deriva ve of the natural logarithm func on Deriva ves of other exponen als and logarithms Other exponen als Other logarithms Logarithmic Differen a on The power rule for irra onal powers
  18. 18. Derivatives of Exponentials Fact If f(x) = ax , then f′ (x) = f′ (0)ax .
  19. 19. Derivatives of Exponentials 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 a −a x h x a −1 h = lim = ax · lim = ax · f′ (0). h→0 h h→0 h
  20. 20. The funny limit in the case of e Ques on eh − 1 What is lim ? h→0 h Solu on
  21. 21. The funny limit in the case of e Ques on eh − 1 What is lim ? h→0 h Solu on ( )n 1 Recall e = lim 1 + = lim (1 + h)1/h . If h is small enough, n→∞ n h→0 e ≈ (1 + h) . So 1/h [ ]h eh − 1 (1 + h)1/h − 1 (1 + h) − 1 h ≈ = = =1 h h h h
  22. 22. The funny limit in the case of e Ques on eh − 1 What is lim ? h→0 h Solu on So in the limit we get equality: eh − 1 lim =1 h→0 h
  23. 23. Derivative of the naturalexponential 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
  24. 24. Exponential Growth Commonly misused term to say something grows exponen ally It means the rate of change (deriva ve) is propor onal to the current value Examples: Natural popula on growth, compounded interest, social networks
  25. 25. Examples Example d Find e3x . dx
  26. 26. Examples Example d Find e3x . dx Solu on d 3x d e = e3x (3x) = 3e3x dx dx
  27. 27. Examples Example d 2 Find ex . dx
  28. 28. Examples Example d 2 Find ex . dx Solu on d x2 2 d 2 e = ex (x2 ) = 2xex dx dx
  29. 29. Examples Example d Find x2 ex . dx
  30. 30. Examples Example d Find x2 ex . dx Solu on d 2 x x e = 2xex + x2 ex dx
  31. 31. Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on Exponen al Growth Deriva ve of the natural logarithm func on Deriva ves of other exponen als and logarithms Other exponen als Other logarithms Logarithmic Differen a on The power rule for irra onal powers
  32. 32. Derivative ofy the natural logarithm Let y = ln x. Then x = e so
  33. 33. Derivative ofy the natural logarithm Let y = ln x. Then x = e so dy ey =1 dx
  34. 34. Derivative ofy the natural logarithm Let y = ln x. Then x = e so dy ey =1 dx dy 1 1 =⇒ = y= dx e x
  35. 35. Derivative ofy the natural logarithm Let y = ln x. Then x = e so dy ey =1 dx dy 1 1 =⇒ = y= dx e x We have discovered: Fact d 1 ln x = dx x
  36. 36. Derivative ofy the natural logarithm Let y = ln x. Then x = e so y dy ey =1 dx dy 1 1 =⇒ = y= ln x dx e x We have discovered: . x Fact d 1 ln x = dx x
  37. 37. Derivative ofy the natural logarithm Let y = ln x. Then x = e so y dy ey =1 dx dy 1 1 =⇒ = y= ln x 1 dx e x x We have discovered: . x Fact d 1 ln x = dx x
  38. 38. The Tower of Powers y y′ x3 3x2 x2 2x1 The deriva ve of a power func on is a power func on of one lower power x1 1x0 x0 0 ? ? x−1 −1x−2 x−2 −2x−3
  39. 39. The Tower of Powers y y′ x3 3x2 x2 2x1 The deriva ve of a power func on is a power func on of one lower power x1 1x0 Each power func on is the deriva ve of x0 0 another power func on, except x−1 ? x−1 x−1 −1x−2 x−2 −2x−3
  40. 40. The Tower of Powers y y′ x3 3x2 x2 2x1 The deriva ve of a power func on is a power func on of one lower power x1 1x0 Each power func on is the deriva ve of x0 0 another power func on, except x−1 ln x x−1 ln x fills in this gap precisely. x−1 −1x−2 x−2 −2x−3
  41. 41. Examples Examples Find deriva ves of these func ons: ln(3x) x ln x √ ln x
  42. 42. Examples Example d Find ln(3x). dx
  43. 43. Examples Example d Find ln(3x). dx Solu on (chain rule way) d 1 1 ln(3x) = ·3= dx 3x x
  44. 44. Examples Example d Find ln(3x). dx Solu on (proper es of logarithms way) d d 1 1 ln(3x) = (ln(3) + ln(x)) = 0 + = dx dx x x The first answer might be surprising un l you see the second solu on.
  45. 45. Examples Example d Find x ln x dx
  46. 46. Examples Example d Find x ln x dx Solu on The product rule is in play here: ( ) ( ) d d d 1 x ln x = x ln x + x ln x = 1 · ln x + x · = ln x + 1 dx dx dx x
  47. 47. Examples Example d √ Find ln x. dx
  48. 48. Examples Example d √ Find ln x. dx Solu on (chain rule way) d √ 1 d√ 1 1 1 ln x = √ x=√ √ = dx x dx x 2 x 2x
  49. 49. Examples Example d √ Find ln x. dx Solu on (proper es of logarithms way) ( ) d √ d 1 1d 1 1 ln x = ln x = ln x = · dx dx 2 2 dx 2 x The first answer might be surprising un l you see the second solu on.
  50. 50. Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on Exponen al Growth Deriva ve of the natural logarithm func on Deriva ves of other exponen als and logarithms Other exponen als Other logarithms Logarithmic Differen a on The power rule for irra onal powers
  51. 51. Other logarithms Example d x Use implicit differen a on to find a. dx
  52. 52. Other logarithms Example d x Use implicit differen a on to find a. dx Solu on Let y = ax , so ln y = ln ax = x ln a
  53. 53. Other logarithms Example d x Use implicit differen a on to find a. dx Solu on Let y = ax , so ln y = ln ax = x ln a Differen ate implicitly: 1 dy dy = ln a =⇒ = (ln a)y = (ln a)ax y dx dx
  54. 54. The funny limit in the case of a x ′ ′ Let y = e . Before we showed y = y (0)y, and now we know y′ = (ln a)y. So Corollary ah − 1 lim = ln a h→0 h In par cular 2h − 1 3h − 1 ln 2 = lim ≈ 0.693 ln 3 = lim ≈ 1.10 h→0 h h→0 h
  55. 55. Other logarithms Example d Find log x. dx a
  56. 56. Other logarithms Example d Find log x. dx a Solu on Let y = loga x, so ay = x.
  57. 57. Other logarithms Example d Find log x. dx a Solu on Let y = loga x, so ay = x. Now differen ate implicitly: dy dy 1 1 (ln a)ay = 1 =⇒ = y = dx dx a ln a x ln a
  58. 58. Other logarithms Example d Find log x. dx a Solu on Or we can use the change of base formula: ln x dy 1 1 y= =⇒ = ln a dx ln a x
  59. 59. More examples Example d Find log2 (x2 + 1) dx
  60. 60. More examples Example d Find log2 (x2 + 1) dx Answer dy 1 1 2x = (2x) = dx ln 2 x2 + 1 (ln 2)(x2 + 1)
  61. 61. Outline Recall Sec on 3.1–3.2 Deriva ve of the natural exponen al func on Exponen al Growth Deriva ve of the natural logarithm func on Deriva ves of other exponen als and logarithms Other exponen als Other logarithms Logarithmic Differen a on The power rule for irra onal powers
  62. 62. A nasty derivative Example √ (x2 + 1) x + 3 Let y = . Find y′ . x−1
  63. 63. A nasty derivative Example √ (x2 + 1) x + 3 Let y = . Find y′ . x−1 Solu on We use the quo ent 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) y′ = 2 (x − 1)2 √ √ 2x x + 3 (x2 + 1) (x2 + 1) x + 3 = + √ − (x − 1) 2 x + 3(x − 1) (x − 1)2
  64. 64. Another way √ (x2 + 1) x + 3 y= x−1
  65. 65. Another way √ (x2 + 1) x + 3 y= x−1 1 ln y = ln(x2 + 1) + ln(x + 3) − ln(x − 1) 2
  66. 66. 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
  67. 67. 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
  68. 68. 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 (x2 + 1) x + 3 = + − dx x 2+1 2(x + 3) x − 1 x−1
  69. 69. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1)
  70. 70. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  71. 71. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  72. 72. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  73. 73. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  74. 74. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  75. 75. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  76. 76. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  77. 77. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  78. 78. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  79. 79. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  80. 80. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  81. 81. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same?
  82. 82. Compare and contrast Using the product, quo ent, 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 differen a on: ( ) 2 √ ′ 2x 1 1 (x + 1) x + 3 y = + − x2 + 1 2(x + 3) x − 1 (x − 1) Are these the same? Yes.
  83. 83. Derivatives of powers Ques on y Let y = xx . Which of these is true? (A) Since y is a power func on, y′ = x · xx−1 = xx . (B) Since y is an exponen al 1 func on, y′ = (ln x) · xx . (C) Neither x 1
  84. 84. Derivatives of powers Ques on y Let y = xx . Which of these is true? (A) Since y is a power func on, y′ = x · xx−1 = xx . (B) Since y is an exponen al 1 func on, y′ = (ln x) · xx . (C) Neither x 1
  85. 85. Why not? Answer y (A) y′ ̸= xx because xx > 0 for all x > 0, and this func on decreases at some places 1 . x 1
  86. 86. Why not? Answer y (A) y′ ̸= xx because xx > 0 for all x > 0, and this func on decreases at some places (B) y′ ̸= (ln x)xx because (ln x)xx = 0 when x = 1, and this func on 1 does not have a horizontal . tangent at x = 1. x 1
  87. 87. It’s neither! Solu on If y = xx , then
  88. 88. It’s neither! Solu on If y = xx , then ln y = x ln x
  89. 89. It’s neither! Solu on If y = xx , then ln y = x ln x 1 dy 1 = x · + ln x = 1 + ln x y dx x
  90. 90. It’s neither! Solu on If y = xx , then ln y = x ln x 1 dy 1 = x · + ln x = 1 + ln x y dx x dy = (1 + ln x)xx = xx + (ln x)xx dx
  91. 91. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx 1 . x 1
  92. 92. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . x 1
  93. 93. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . x 1
  94. 94. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . x 1
  95. 95. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . x 1
  96. 96. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . y′ < 0 on the interval (0, e−1 ) x 1
  97. 97. Or both? Solu on d x y x = xx + (ln x)xx = (1 + ln x)xx dx Remarks Each of these terms is one of the 1 wrong answers! . y′ < 0 on the interval (0, e−1 ) x 1 y′ = 0 when x = e−1
  98. 98. Derivatives of power functionswith any exponent Fact (The power rule) Let y = xr . Then y′ = rxr−1 .
  99. 99. Derivatives of power functionswith any exponent Fact (The power rule) Let y = xr . Then y′ = rxr−1 . Proof. y = xr =⇒ ln y = r ln x Now differen ate: 1 dy r dy y = =⇒ = r = rxr−1 y dx x dx x
  100. 100. Summary Deriva ves of y y′ Logarithmic and Exponen al Func ons ex ex Logarithmic Differen a on can allow ax (ln a) · ax us to avoid the product 1 and quo ent rules. ln x x We are finally done with 1 1 loga x · the Power Rule! ln a x

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