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- 1. Section 2.6 Implicit Differentiation V63.0121.006/016, Calculus I February 23, 2010 Announcements Quiz 2 is February 26, covering §§1.5–2.3 Midterm is March 4, covering §§1.1–2.5 On HW 5, Problem 2.3.46 should be 2.4.46 . . Image credit: Telstar Logistics . . . . . .
- 2. Announcements on white background Announcements Quiz 2 is February 26, covering §§1.5–2.3 Midterm is March 4, covering §§1.1–2.5 On HW 5, Problem 2.3.46 should be 2.4.46 . . . . . .
- 3. Outline The big idea, by example Examples Basic Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
- 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. 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. 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). . . . . . . .
- 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 −?) . . . . . .
- 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 . . . . . .
- 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. 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 . . . . . .
- 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 . . . . . .
- 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 . . . . . .
- 13. 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) . . . . . .
- 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. 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 . . . . . . . .
- 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 . . l .ooks like a function . . . . . .
- 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. 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 . . . . . . .
- 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. l .ooks like a function . x . . . . . . .
- 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. 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 . . . . . . .
- 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. . . x . . does not look like a function, but that’s OK—there are only two points like this . . . . . .
- 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 . l .ooks like a function . . . . . .
- 24. 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 . . . . . .
- 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). . . . . . .
- 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 . . . . . .
- 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! . . . . . .
- 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 . . . . . .
- 29. 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 . . . . . .
- 30. 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. . . . . . .
- 31. Outline The big idea, by example Examples Basic Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
- 32. Example Find y′ along the curve y3 + 4xy = x2 + 3. . . . . . .
- 33. Example Find y′ along the curve y3 + 4xy = x2 + 3. Solution Implicitly differentiating, we have 3y2 y′ + 4(1 · y + x · y′ ) = 2x . . . . . .
- 34. Example Find y′ along the curve y3 + 4xy = x2 + 3. Solution Implicitly differentiating, we have 3y2 y′ + 4(1 · y + x · y′ ) = 2x Solving for y′ gives 3y2 y′ + 4xy′ = 2x − 4y (3y2 + 4x)y′ = 2x − 4y 2x − 4y =⇒ y′ = 2 3y + 4x . . . . . .
- 35. Example Find y′ if y5 + x2 y3 = 1 + y sin(x2 ). . . . . . .
- 36. 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 . . . . . .
- 37. 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). . . . . . . .
- 38. 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 . . . . . .
- 39. 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 . . . . . .
- 40. 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. . . . . . .
- 41. Example Find the horizontal tangent lines to the same curve: y2 = x3 + x2 . . . . . .
- 42. 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] . . . . . .
- 43. 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. . . . . . .
- 44. 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. . . . . . .
- 45. 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 3 2 2 4 2 y = − + − = =⇒ y = ± √ , 3 3 27 3 3 so there are two horizontal tangents. . . . . . .
- 46. Horizontal Tangents ( ) . − 2 , 3√3 3 2 . . . ( ) . − 2 , − 3 √3 3 2 . . . . . .
- 47. Horizontal Tangents ( ) . − 2 , 3√3 3 2 . . . ( ) . − 2 , − 3 √3 3 2 n . ode . . . . . .
- 48. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 . . . . . .
- 49. Example Find the vertical tangent lines to the same curve: y2 = x3 + x2 Solution dx Tangent lines are vertical when = 0. dy . . . . . .
- 50. 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). . . . . . .
- 51. 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] . . . . . .
- 52. Solution, continued Solving the second equation gives 2y = 0 =⇒ 2y = 0 =⇒ y = 0 3x2 + 2x (as long as 3x2 + 2x ̸= 0). . . . . . .
- 53. 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. . . . . . .
- 54. 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. . . . . . .
- 55. Tangents ( ) . − 2 , 3√3 3 2 . . −1 , 0 ) . ( . . ( ) . − 2 , − 3 √3 3 2 n . ode . . . . . .
- 56. Examples Example Show that the families of curves xy = c x2 − y2 = k are orthogonal, that is, they intersect at right angles. . . . . . .
- 57. Orthogonal Families of Curves y . xy = c x2 − y2 = k . x . . . . . . .
- 58. Orthogonal Families of Curves y . .xy = 1 xy = c x2 − y2 = k . x . . . . . . .
- 59. Orthogonal Families of Curves y . .xy = .xy 2 = 1 xy = c x2 − y2 = k . x . . . . . . .
- 60. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 xy = c x2 − y2 = k . x . . . . . . .
- 61. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 xy = c x2 − y2 = k . x . 1 − = .xy . . . . . .
- 62. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 xy = c x2 − y2 = k . x . 1 − 2 = − .xy = .xy . . . . . .
- 63. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = 1 xy = c x2 − y2 = k . x . 1 − − 2 = − .xy 3 = .xy = .xy . . . . . .
- 64. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − − 2 = x − .xy 3 = .xy = .xy . . . . . .
- 65. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 2 . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − − 2 = x x − .xy 3 = .xy = .xy . . . . . .
- 66. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 3 . − y2 = 2 . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − x2 − 2 = x x − .xy 3 = .xy = .xy . . . . . .
- 67. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 3 . − y2 = 2 . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − x2 . 2 − y2 = −1 − 2 x = x x − .xy 3 = .xy = .xy . . . . . .
- 68. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 3 . − y2 = 2 . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − x2 . 2 − y2 = −1 − 2 x = x x − .xy 3 . 2 − y2 = −2 x = .xy = .xy . . . . . .
- 69. Orthogonal Families of Curves y . .xy .xy = 3 = .xy 2 = . 2 − y2 = 3 . − y2 = 2 . 2 − y2 = 1 1 xy = c x2 − y2 = k . x . 1 − x2 . 2 − y2 = −1 − 2 x = x x − .xy 3 . 2 − y2 = −2 x2 = . − y2 = −3 .xy x = .xy . . . . . .
- 70. 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 . . . . . .
- 71. 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. . . . . . .
- 72. Music Selection “The Curse of Curves” by Cute is What We Aim For . . . . . .
- 73. 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 . . . . . .
- 74. Compressibility Deﬁnition The isothermic compressibility of a ﬂuid is deﬁned by dV 1 β=− dP V with temperature held constant. . . . . . .
- 75. 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. . . . . . .
- 76. Example Find the isothermic compressibility of an ideal gas. . . . . . .
- 77. 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. . . . . . .
- 78. 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. . . . . . .
- 79. 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 size. . . Image credit: Wikimedia Commons . . . . . .
- 80. 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 . . . . . .
- 81. 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 . . . . . .
- 82. 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? . . . . . .
- 83. 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 . . . . . .
- 84. 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 . . . . . .
- 85. 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). . . . . . .
- 86. Outline The big idea, by example Examples Basic Examples Vertical and Horizontal Tangents Orthogonal Trajectories Chemistry The power rule for rational powers . . . . . .
- 87. Using implicit differentiation to ﬁnd derivatives Example dy √ Find if y = x. dx . . . . . .
- 88. 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 . . . . . .
- 89. 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 . . . . . .
- 90. 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. First, raise both sides to the qth power: y = xp/q =⇒ yq = xp . . . . . .
- 91. 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. First, raise both sides to the qth power: y = xp/q =⇒ yq = xp Now, differentiate implicitly: dy dy p xp−1 qyq−1 = pxp−1 =⇒ = · q −1 dx dx q y . . . . . .
- 92. 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. First, raise both sides to the qth power: y = xp/q =⇒ yq = xp Now, differentiate implicitly: dy dy p xp−1 qyq−1 = pxp−1 =⇒ = · q −1 dx dx q y Simplify: yq−1 = x(p/q)(q−1) = xp−p/q so x p −1 xp−1 = p−p/q = xp−1−(p−p/q) = xp/q−1 y q −1 x . . . . . .
- 93. What have we learned today? Implicit Differentiation allows us to pretend that a relation describes a function, since it does, locally, “almost everywhere.” The Power Rule was established for powers which are rational numbers. . . . . . .

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