61 d fy withtionships.#r20410-12X30 bZ40200-20--24020z 0-20--40-2-1-1yy0011fxy=fyx2210-12x2210-12XFiets40² 20fxyy = (fxy)y=0-2=4020ZO-20-40--2-1-10y0yfyfyy112210-12XухNotice that fxy = fyx in Example 7. This is not just a coincidence. It turns out that themixed partial derivatives fxy and fyx are equal for most functions that one meets in prac-tice. The following theorem, which was discovered by the French mathematician AlexisClairaut (1713-1765), gives conditions under which we can assert that fxy = fyx. Theproof is given in Appendix F.2210-1²-2XClairaut's Theorem Suppose f is defined on a disk D that contains the point(a, b). If the functions fxy and fyx are both continuous on D, thenfxy(a, b) = fyx(a, b)Partial derivatives of order 3 or higher can also be defined. For instance,aa³fay (ayax) - ay ² axдуду ander hand-andthem.dr42. f(x, y) = y sin(xy); f(1, 1)43. f(x, y, z) = In144. f(x, y, z) = x; f(e, 1, 0)y2x + 3y57. v = sin(s² - 1²)155. z =16012960 0907 yil45-46 Use the definition of partial derivatives as limits (4) to findf(x, y) and f(x, y).45. f(x, y) = xy² – x³y+51-52 Find az/ax and az/ay.51. (a) z = f(x) + g(y)52. (a) z = f(x) g(y)(c) z = f(x/y)69. w =2/x² + y² + z²47-50 Use implicit differentiation to find az/ax and az/ay.247. x² + 2y² + 3z² = 149. e² = xyzX/x² + y² + z²68. V = In (r + s² + 1³);y + 2znel 1653-58 Find all the second partial derivatives.53. f(x, y) = xy - 2x³y²a ³Wdu² dv-; (1, 2, 2)J³wəz ay əx46. f(x, y)a³Var as at59-62 Verify that the conclusion of Clairaut's Theorem holds, thatis, Uxy = Uyx.SVIJ 16059. u = x²y³ - y461. u = cos(x²y)63-70 Find the indicated partial derivative(s).63. f(x, y) = x4y² – x³y; fxxx, fxyx64. f(x, y) = sin(2x + 5y); fyxy65. f(x, y, z) = exyz²; fxyz66. g(r, s, t) = e' sin(st); grst67. W = √√√u + v²;Xx + y²48. x² - y² + z² = 2z = 4obil 107.0826250. yz + x ln y = z²(b) z=f(x + y) w sf.oe(b) z = f(xy)XX54. f(x, y) = ln(ax + by)56. T = e-2r cos 0 brut58. w = √√√1 + uv²2d³ wдх2 ду60. u = exy sin y62. u = ln(x + 2y)vilicoq sAppa70. u = xªy¹zc;SECTION 14.3 Partial Derivatives71. If f(x, y, z) = xy²z³ + arcsin(x√z), find fxzy.[Hint: Which order of differentiation is easiest?]Jouəx əy² əz³noirOU72. If g(x, y, z)=√1 + x₂ + √√1 - xy, find gxyz. [Hint: Use adifferent order of differentiation for each term.]DOYX73. Use the table of values of f(x, y) to estimate the values off.(3, 2), f.(3, 2.2), and fy(3, 2).2.53.03.5y1.812.5YAal tiuosio Iapitolo loll104. I18.120.0(b) fy(e) fyy2.010.2P,17.522.42.29.315.974. Level curves are shown for a function f. Determine whetherthe following partial derivatives are positive or negative at thedipoint P.bre(a) fib-(c) fxx 01150291(d) fxysprbna26.192511/10 8 6 4 2x"X35 - 9 moim rouborq asiguo ddeworl2 AS75. Verify that the function u = e-a²k²t sin kx is a solution of theheat conduction equation u, =a²uxx.AVA76. Determine whether each of the following functions is asolution of Laplace's equation uxx + Uyy = 0.(a) u = x² + y²(b) u = x² - y²(c) u = x³ + 3xy²(e) u = sin x cosh y + cos x sinh y(f) u = e cos y - e cos x(d) u = ln √√x² + y²-77. Verify that the function u = 1/√√x² + y² + z² is a solution ofthe three-dimensional Laplace equation Uxx + Uyy + Uzz= 0.=78. Show that each of the following functions is a solution of thewave equation u₁t = a²uxx.(a) u sin(kx) sin(akt)(c) u = (x - at) + (x + at)6=(b) u = t/(a²t² - x²)(d) u = sin(x - at) + In(x + at)79. If f and g are twice differentiable functions of a single vari-able, show that the functionu(x, t) = f(x + at) + g(x - at)is a solution of the wave equation given in Exercise 78.

Question

61 d fy withtionships.#r20410-12X30 bZ40200-20--24020z 0-20--40-2-1-1yy0011fxy=fyx2210-12x2210-12XFiets40² 20fxyy = (fxy)y=0-2=4020ZO-20-40--2-1-10y0yfyfyy112210-12XухNotice that fxy = fyx in Example 7. This is not just a coincidence. It turns out that themixed partial derivatives fxy and fyx are equal for most functions that one meets in prac-tice. The following theorem, which was discovered by the French mathematician AlexisClairaut (1713-1765), gives conditions under which we can assert that fxy = fyx. Theproof is given in Appendix F.2210-1²-2XClairaut's Theorem Suppose f is defined on a disk D that contains the point(a, b). If the functions fxy and fyx are both continuous on D, thenfxy(a, b) = fyx(a, b)Partial derivatives of order 3 or higher can also be defined. For instance,aa³fay (ayax) - ay ² axдуду ander hand-andthem.dr42. f(x, y) = y sin(xy); f(1, 1)43. f(x, y, z) = In144. f(x, y, z) = x; f(e, 1, 0)y2x + 3y57. v = sin(s² - 1²)155. z =16012960 0907 yil45-46 Use the definition of partial derivatives as limits (4) to findf(x, y) and f(x, y).45. f(x, y) = xy² – x³y+51-52 Find az/ax and az/ay.51. (a) z = f(x) + g(y)52. (a) z = f(x) g(y)(c) z = f(x/y)69. w =2/x² + y² + z²47-50 Use implicit differentiation to find az/ax and az/ay.247. x² + 2y² + 3z² = 149. e² = xyzX/x² + y² + z²68. V = In (r + s² + 1³);y + 2znel 1653-58 Find all the second partial derivatives.53. f(x, y) = xy - 2x³y²a ³Wdu² dv-; (1, 2, 2)J³wəz ay əx46. f(x, y)a³Var as at59-62 Verify that the conclusion of Clairaut's Theorem holds, thatis, Uxy = Uyx.SVIJ 16059. u = x²y³ - y461. u = cos(x²y)63-70 Find the indicated partial derivative(s).63. f(x, y) = x4y² – x³y; fxxx, fxyx64. f(x, y) = sin(2x + 5y); fyxy65. f(x, y, z) = exyz²; fxyz66. g(r, s, t) = e' sin(st); grst67. W = √√√u + v²;Xx + y²48. x² - y² + z² = 2z = 4obil 107.0826250. yz + x ln y = z²(b) z=f(x + y) w sf.oe(b) z = f(xy)XX54. f(x, y) = ln(ax + by)56. T = e-2r cos 0 brut58. w = √√√1 + uv²2d³ wдх2 ду60. u = exy sin y62. u = ln(x + 2y)vilicoq sAppa70. u = xªy¹zc;SECTION 14.3 Partial Derivatives71. If f(x, y, z) = xy²z³ + arcsin(x√z), find fxzy.[Hint: Which order of differentiation is easiest?]Jouəx əy² əz³noirOU72. If g(x, y, z)=√1 + x₂ + √√1 - xy, find gxyz. [Hint: Use adifferent order of differentiation for each term.]DOYX73. Use the table of values of f(x, y) to estimate the values off.(3, 2), f.(3, 2.2), and fy(3, 2).2.53.03.5y1.812.5YAal tiuosio Iapitolo loll104. I18.120.0(b) fy(e) fyy2.010.2P,17.522.42.29.315.974. Level curves are shown for a function f. Determine whetherthe following partial derivatives are positive or negative at thedipoint P.bre(a) fib-(c) fxx 01150291(d) fxysprbna26.192511/10 8 6 4 2x&#34;X35 - 9 moim rouborq asiguo ddeworl2 AS75. Verify that the function u = e-a²k²t sin kx is a solution of theheat conduction equation u, =a²uxx.AVA76. Determine whether each of the following functions is asolution of Laplace's equation uxx + Uyy = 0.(a) u = x² + y²(b) u = x² - y²(c) u = x³ + 3xy²(e) u = sin x cosh y + cos x sinh y(f) u = e cos y - e cos x(d) u = ln √√x² + y²-77. Verify that the function u = 1/√√x² + y² + z² is a solution ofthe three-dimensional Laplace equation Uxx + Uyy + Uzz= 0.=78. Show that each of the following functions is a solution of thewave equation u₁t = a²uxx.(a) u sin(kx) sin(akt)(c) u = (x - at) + (x + at)6=(b) u = t/(a²t² - x²)(d) u = sin(x - at) + In(x + at)79. If f and g are twice differentiable functions of a single vari-able, show that the functionu(x, t) = f(x + at) + g(x - at)is a solution of the wave equation given in Exercise 78.

Accepted Answer

61.
The given function u=cosx2y.
We have to verify that uxy=uyx.