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# Lesson 11: Implicit Differentiation

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Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.

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### Lesson 11: Implicit Differentiation

1. 1. Section 2.6 Implicit Differentiation V63.0121.027, Calculus I October 8, 2009 Announcements Midterm next Thursday, covering §§1.1–2.4. . . Image credit: Telstar Logistics . . . . . .
2. 2. Outline The big idea, by example Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
3. 3. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . . . . . .
4. 4. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . . . . . .
5. 5. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . . . . . . .
6. 6. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . Solution (Explicit) √ Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?) . . . . . .
7. 7. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . Solution (Explicit) √ Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?) dy −2x x Differentiate: =− √ =√ dx 2 1−x 2 1 − x2 . . . . . .
8. 8. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . Solution (Explicit) √ Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?) dy −2x x Differentiate: =− √ =√ dx 2 1−x 2 1 − x2 dy 3 /5 3/5 3 Evaluate: =√ = = . dx x=3/5 1 − (3 /5 )2 4/5 4 . . . . . .
9. 9. Motivating Example y . Problem Find the slope of the line which is tangent to the curve . x . 2 2 x +y =1 at the point (3/5, −4/5). . Solution (Explicit) √ Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?) dy −2x x Differentiate: =− √ =√ dx 2 1−x 2 1 − x2 dy 3 /5 3/5 3 Evaluate: =√ = = . dx x=3/5 1 − (3 /5 )2 4/5 4 . . . . . .
10. 10. Motivating Example, another way We know that x2 + y2 = 1 does not deﬁne y as a function of x, but suppose it did. Suppose we had y = f(x), so that x2 + (f(x))2 = 1 . . . . . .
11. 11. Motivating Example, another way We know that x2 + y2 = 1 does not deﬁne y as a function of x, but suppose it did. Suppose we had y = f(x), so that x2 + (f(x))2 = 1 We could differentiate this equation to get 2x + 2f(x) · f′ (x) = 0 . . . . . .
12. 12. Motivating Example, another way We know that x2 + y2 = 1 does not deﬁne y as a function of x, but suppose it did. Suppose we had y = f(x), so that x2 + (f(x))2 = 1 We could differentiate this equation to get 2x + 2f(x) · f′ (x) = 0 We could then solve to get x f ′ (x ) = − f(x) . . . . . .
13. 13. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . x . . . . . . . .
14. 14. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . x . . . . . . . .
15. 15. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . x . . l .ooks like a function . . . . . .
16. 16. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the . graph of a function. . x . . . . . . .
17. 17. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the . graph of a function. . x . . . . . . .
18. 18. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the . graph of a function. l .ooks like a function . x . . . . . . .
19. 19. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . . x . . . . . . .
20. 20. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . . x . . . . . . .
21. 21. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. . . x . . does not look like a function, but that’s OK—there are only two points like this . . . . . .
22. 22. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. So f(x) is deﬁned “locally”, almost . x . everywhere and is differentiable . l .ooks like a function . . . . . .
23. 23. Yes, we can! The beautiful fact (i.e., deep theorem) is that this works! “Near” most points on y . the curve x2 + y2 = 1, the curve resembles the graph of a function. So f(x) is deﬁned “locally”, almost . x . everywhere and is differentiable The chain rule then . applies for this local choice. l .ooks like a function . . . . . .
24. 24. Motivating Example, again, with Leibniz notation Problem Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the point (3/5, −4/5). . . . . . .
25. 25. Motivating Example, again, with Leibniz notation Problem Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the point (3/5, −4/5). Solution dy Differentiate: 2x + 2y =0 dx . . . . . .
26. 26. Motivating Example, again, with Leibniz notation Problem Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the point (3/5, −4/5). Solution dy Differentiate: 2x + 2y =0 dx Remember y is assumed to be a function of x! . . . . . .
27. 27. Motivating Example, again, with Leibniz notation Problem Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the point (3/5, −4/5). Solution dy Differentiate: 2x + 2y =0 dx Remember y is assumed to be a function of x! dy x Isolate: =− . dx y . . . . . .
28. 28. Motivating Example, again, with Leibniz notation Problem Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the point (3/5, −4/5). Solution dy Differentiate: 2x + 2y =0 dx Remember y is assumed to be a function of x! dy x Isolate: =− . dx y dy 3 /5 3 Evaluate: = = . dx ( 3 ,− 4 ) 4/5 4 5 5 . . . . . .
29. 29. Summary If a relation is given between x and y which isn’t a function: “Most of the time”, i.e., “at most places” y can be y . assumed to be a function of . x we may differentiate the . x . relation as is dy Solving for does give the dx slope of the tangent line to the curve at a point on the curve. . . . . . .
30. 30. Mnemonic Explicit Implicit y = f(x) F(x, y) = k . . . . . .
31. 31. Outline The big idea, by example Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
32. 32. Example Find the equation of the line tangent to the curve . y 2 = x 2 (x + 1 ) = x 3 + x 2 at the point (3, −6). . . . . . . .
33. 33. Example Find the equation of the line tangent to the curve . y 2 = x 2 (x + 1 ) = x 3 + x 2 at the point (3, −6). . Solution Differentiating the expression implicitly with respect to x gives dy dy 3x2 + 2x 2y = 3x2 + 2x, so = , and dx dx 2y dy 3 · 32 + 2 · 3 33 11 = =− =− . dx (3,−6) 2(−6) 12 4 . . . . . .
34. 34. Example Find the equation of the line tangent to the curve . y 2 = x 2 (x + 1 ) = x 3 + x 2 at the point (3, −6). . Solution Differentiating the expression implicitly with respect to x gives dy dy 3x2 + 2x 2y = 3x2 + 2x, so = , and dx dx 2y dy 3 · 32 + 2 · 3 33 11 = =− =− . dx (3,−6) 2(−6) 12 4 11 Thus the equation of the tangent line is y + 6 = − (x − 3). 4 . . . . . .
35. 35. Line equation forms slope-intercept form y = mx + b where the slope is m and (0, b) is on the line. point-slope form y − y0 = m(x − x0 ) where the slope is m and (x0 , y0 ) is on the line. . . . . . .
36. 36. Example Find the horizontal tangent lines to the same curve: y2 = x3 + x2 . . . . . .
37. 37. Example Find the horizontal tangent lines to the same curve: y2 = x3 + x2 Solution We have to solve these two equations: . . 2 3 2 3x2 + 2x y = x +x = 0 1 . [(x, y). is on the curve] 2 . 2y [tangent line is horizontal] . . . . . .
38. 38. Solution, continued Solving the second equation gives 3x2 + 2x = 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0 2y (as long as y ̸= 0). So x = 0 or 3x + 2 = 0. . . . . . .
39. 39. Solution, continued Solving the second equation gives 3x2 + 2x = 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0 2y (as long as y ̸= 0). So x = 0 or 3x + 2 = 0. Substituting x = 0 into the ﬁrst equation gives y2 = 03 + 02 = 0 =⇒ y = 0 which we’ve disallowed. So no horizontal tangents down that road. . . . . . .
40. 40. Solution, continued Solving the second equation gives 3x2 + 2x = 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0 2y (as long as y ̸= 0). So x = 0 or 3x + 2 = 0. Substituting x = 0 into the ﬁrst equation gives y2 = 03 + 02 = 0 =⇒ y = 0 which we’ve disallowed. So no horizontal tangents down that road. Substituting x = −2/3 into the ﬁrst equation gives 2 2 2 y2 = (− )3 + (− )2 =⇒ y = ± √ , 3 3 3 3 so there are two horizontal tangents. . . . . . .
41. 41. Horizontal Tangents ( ) . − 2 , 3√3 3 2 . . . ( ) . − 2 , − 3 √3 3 2 . . . . . .
42. 42. Horizontal Tangents ( ) . − 2 , 3√3 3 2 . . . ( ) . − 2 , − 3 √3 3 2 n . ode . . . . . .
43. 43. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 . . . . . .
44. 44. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 Solution dx Tangent lines are vertical when = 0. dy . . . . . .
45. 45. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 Solution dx Tangent lines are vertical when = 0. dy Differentiating x implicitly as a function of y gives dx dx dx 2y 2y = 3x2 + 2x , so = 2 (notice this is the dy dy dy 3x + 2x reciprocal of dy/dx). . . . . . .
46. 46. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 Solution dx Tangent lines are vertical when = 0. dy Differentiating x implicitly as a function of y gives dx dx dx 2y 2y = 3x2 + 2x , so = 2 (notice this is the dy dy dy 3x + 2x reciprocal of dy/dx). We must solve . . 2y y2 = x 3 + x2 =0 1 . [(x, y). is on the curve] 2 . 3x2 + 2x [tangent line is vertical] . . . . . .
47. 47. Solution, continued Solving the second equation gives 2y = 0 =⇒ 2y = 0 =⇒ y = 0 3x2 + 2x (as long as 3x2 + 2x ̸= 0). . . . . . .
48. 48. Solution, continued Solving the second equation gives 2y = 0 =⇒ 2y = 0 =⇒ y = 0 3x2 + 2x (as long as 3x2 + 2x ̸= 0). Substituting y = 0 into the ﬁrst equation gives 0 = x3 + x2 = x2 (x + 1) So x = 0 or x = −1. . . . . . .
49. 49. Solution, continued Solving the second equation gives 2y = 0 =⇒ 2y = 0 =⇒ y = 0 3x2 + 2x (as long as 3x2 + 2x ̸= 0). Substituting y = 0 into the ﬁrst equation gives 0 = x3 + x2 = x2 (x + 1) So x = 0 or x = −1. x = 0 is not allowed by the ﬁrst equation, but dx = 0, dy (−1,0) so here is a vertical tangent. . . . . . .
50. 50. Vertical Tangents . −1 , 0 ) . ( . n . ode . . . . . .
51. 51. Example Find y′ if y5 + x2 y3 = 1 + y sin(x2 ). . . . . . .
52. 52. Example Find y′ if y5 + x2 y3 = 1 + y sin(x2 ). Solution Differentiating implicitly: 5y4 y′ + (2x)y3 + x2 (3y2 y′ ) = y′ sin(x2 ) + y cos(x2 )(2x) Collect all terms with y′ on one side and all terms without y′ on the other: 5y4 y′ + 3x2 y2 y′ − sin(x2 )y′ = −2xy3 + 2xy cos(x2 ) Now factor and divide: 2xy(cos x2 − y2 ) y′ = 5y4 + 3x2 y2 − sin x2 . . . . . .
53. 53. Examples Example Show that the families of curves xy = c x2 − y2 = k are orthogonal, that is, they intersect at right angles. . . . . . .
54. 54. Orthogonal Families of Curves y . . x . . . . . . .
55. 55. Orthogonal Families of Curves y . .xy = 1 . x . . . . . . .
56. 56. Orthogonal Families of Curves y . .xy = .xy 2 = 1 . x . . . . . . .
57. 57. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 . x . . . . . . .
58. 58. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 . x . 1 − = .xy . . . . . .
59. 59. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 . x . 1 − 2 = − .xy = .xy . . . . . .
60. 60. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 . x . 1 − − 2 = − .xy 3 = .xy = .xy . . . . . .
61. 61. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 1 1 . x . 1 − − 2 = x − .xy 3 = .xy = .xy . . . . . .
62. 62. x . 2 − y2 = 2 x . 2 − y2 = 1 . . y Orthogonal Families of Curves .xy .xy = = − .xy .xy 1 = = − .xy 1 = − 2 . .xy 3 2 = 3 . . . . x . .
63. 63. . 2 − y2 = 3 x2 x . − y2 = 2 x . 2 − y2 = 1 . . y Orthogonal Families of Curves .xy .xy = = − .xy .xy 1 = = − .xy 1 = − 2 . .xy 3 2 = 3 . . . . x . .
64. 64. Examples Example Show that the families of curves xy = c x2 − y2 = k are orthogonal, that is, they intersect at right angles. Solution In the ﬁrst curve, y y + xy′ = 0 =⇒ y′ = − x . . . . . .
65. 65. Examples Example Show that the families of curves xy = c x2 − y2 = k are orthogonal, that is, they intersect at right angles. Solution In the ﬁrst curve, y y + xy′ = 0 =⇒ y′ = − x In the second curve, x 2x − 2yy′ = 0 = =⇒ y′ = y The product is −1, so the tangent lines are perpendicular wherever they intersect. . . . . . .
66. 66. Music Selection “The Curse of Curves” by Cute is What We Aim For . . . . . .
67. 67. Ideal gases The ideal gas law relates temperature, pressure, and volume of a gas: PV = nRT (R is a constant, n is the amount of gas in moles) . . Image credit: Scott Beale / Laughing Squid . . . . . .
68. 68. Compressibility Deﬁnition The isothermic compressibility of a ﬂuid is deﬁned by dV 1 β=− dP V with temperature held constant. . . . . . .
69. 69. Compressibility Deﬁnition The isothermic compressibility of a ﬂuid is deﬁned by dV 1 β=− dP V with temperature held constant. Approximately we have ∆V dV ∆V ≈ = −β V =⇒ ≈ −β∆P ∆P dP V The smaller the β , the “harder” the ﬂuid. . . . . . .
70. 70. Example Find the isothermic compressibility of an ideal gas. . . . . . .
71. 71. Example Find the isothermic compressibility of an ideal gas. Solution If PV = k (n is constant for our purposes, T is constant because of the word isothermic, and R really is constant), then dP dV dV V ·V+P = 0 =⇒ =− dP dP dP P So 1 dV 1 β=− · = V dP P Compressibility and pressure are inversely related. . . . . . .
72. 72. Nonideal gasses Not that there’s anything wrong with that Example The van der Waals equation makes fewer simpliﬁcations: H .. ( ) O . . xygen . . n2 H P + a 2 (V − nb) = nRT, . V H .. where P is the pressure, V the O . . xygen H . ydrogen bonds volume, T the temperature, n H .. the number of moles of the . gas, R a constant, a is a O . . xygen . . H measure of attraction between particles of the gas, H .. and b a measure of particle size. . . . . . .
73. 73. Nonideal gasses Not that there’s anything wrong with that Example The van der Waals equation makes fewer simpliﬁcations: ( ) n2 P + a 2 (V − nb) = nRT, V where P is the pressure, V the volume, T the temperature, n the number of moles of the gas, R a constant, a is a measure of attraction between particles of the gas, and b a measure of particle . Image size. . . . . . . .
74. 74. Let’s ﬁnd the compressibility of a van der Waals gas. Differentiating the van der Waals equation by treating V as a function of P gives ( ) ( ) an2 dV 2an2 dV P+ 2 + (V − bn) 1 − 3 = 0, V dP V dP . . . . . .
75. 75. Let’s ﬁnd the compressibility of a van der Waals gas. Differentiating the van der Waals equation by treating V as a function of P gives ( ) ( ) an2 dV 2an2 dV P+ 2 + (V − bn) 1 − 3 = 0, V dP V dP so 1 dV V2 (V − nb) β=− = V dP 2abn3 − an2 V + PV3 . . . . . .
76. 76. Let’s ﬁnd the compressibility of a van der Waals gas. Differentiating the van der Waals equation by treating V as a function of P gives ( ) ( ) an2 dV 2an2 dV P+ 2 + (V − bn) 1 − 3 = 0, V dP V dP so 1 dV V2 (V − nb) β=− = V dP 2abn3 − an2 V + PV3 What if a = b = 0? . . . . . .
77. 77. Let’s ﬁnd the compressibility of a van der Waals gas. Differentiating the van der Waals equation by treating V as a function of P gives ( ) ( ) an2 dV 2an2 dV P+ 2 + (V − bn) 1 − 3 = 0, V dP V dP so 1 dV V2 (V − nb) β=− = V dP 2abn3 − an2 V + PV3 What if a = b = 0? dβ Without taking the derivative, what is the sign of ? db . . . . . .
78. 78. Let’s ﬁnd the compressibility of a van der Waals gas. Differentiating the van der Waals equation by treating V as a function of P gives ( ) ( ) an2 dV 2an2 dV P+ 2 + (V − bn) 1 − 3 = 0, V dP V dP so 1 dV V2 (V − nb) β=− = V dP 2abn3 − an2 V + PV3 What if a = b = 0? dβ Without taking the derivative, what is the sign of ? db dβ Without taking the derivative, what is the sign of ? da . . . . . .
79. 79. Nasty derivatives dβ (2abn3 − an2 V + PV3 )(nV2 ) − (nbV2 − V3 )(2an3 ) =− db (2abn3 − an2 V + PV3 )2 ( 2 ) nV3 an + PV2 = −( )2 < 0 PV3 + an2 (2bn − V) dβ n2 (bn − V)(2bn − V)V2 = ( )2 > 0 da PV3 + an2 (2bn − V) (as long as V > 2nb, and it’s probably true that V ≫ 2nb). . . . . . .
80. 80. Outline The big idea, by example Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
81. 81. Using implicit differentiation to ﬁnd derivatives Example dy √ Find if y = x. dx . . . . . .
82. 82. Using implicit differentiation to ﬁnd derivatives Example dy √ Find if y = x. dx Solution √ If y = x, then y2 = x, so dy dy 1 1 2y = 1 =⇒ = = √ . dx dx 2y 2 x . . . . . .
83. 83. The power rule for rational powers Theorem p p/q−1 If y = xp/q , where p and q are integers, then y′ = x . q . . . . . .
84. 84. The power rule for rational powers Theorem p p/q−1 If y = xp/q , where p and q are integers, then y′ = x . q Proof. We have dy dy p x p −1 yq = xp =⇒ qyq−1 = pxp−1 =⇒ = · q−1 dx dx q y . . . . . .
85. 85. The power rule for rational powers Theorem p p/q−1 If y = xp/q , where p and q are integers, then y′ = x . q Proof. We have dy dy p x p −1 yq = xp =⇒ qyq−1 = pxp−1 =⇒ = · q−1 dx dx q y Now yq−1 = x(p/q)(q−1) = xp−p/q so x p −1 = xp−1−(p−p/q) = xp/q−1 y q −1 . . . . . .
86. 86. Summary If a relation is given between x and y which isn’t a function: “Most of the time”, i.e., “at most places” y can be y . assumed to be a function of . x we may differentiate the . x . relation as is dy Solving for does give the dx slope of the tangent line to the curve at a point on the curve. . . . . . .