Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 13.3 Derivadas parciales 915
En los ejercicios 59 a 64, calcular las derivadas parciales de
primer orden con respecto a x, y y z.
En los ejercicios 65 a 70, evaluar f x , f y y f z en el punto dado.
En los ejercicios 71 a 80, calcular las cuatro derivadas parciales
de segundo orden. Observar que las derivadas parciales mixtas
de segundo orden son iguales.
En los ejercicios 81 a 88, para f(x, y), encontrar todos los valores
de x y y, tal que f x (x, y) = 0 y f y (x, y) = 0 simultáneamente.
En los ejercicios 89 a 92, utilizar un sistema algebraico por computadora
y hallar las derivadas parciales de primero y segundo
orden de la función. Determinar si existen valores de x y y tales
que y simultáneamente.
En los ejercicios 93 a 96, mostrar que las derivadas parciales
mixtas f xyy , f yxy y f yyx son iguales.
Ecuación de Laplace
En los ejercicios 97 a 100, mostrar que la
función satisface la ecuación de Laplace
97. 98.
99. z e x sen y 100.
Ecuación de ondas
En los ejercicios 101 a 104, mostrar que la
función satisface la ecuación de ondas
101. z sen(x ct) 102.
103. 104. z sen wct sen wx
Ecuación del calor
En los ejercicios 105 y 106, mostrar que la
función satisface la ecuación del calor
105. 106.
En los ejercicios 107 y 108, determinar si existe o no una función
f(x, y) con las derivadas parciales dadas. Explicar el razonamiento.
Si tal función existe, dar un ejemplo.
En los ejercicios 109 y 110, encontrar la primera derivada parcial
con respecto a x.
z e t sin x c
z e t cos x c
z/t c 2 2 z/x 2 .
z lnx ct
z cos4x 4ct
2 z/t 2 c 2 2 z/x 2 .
z arctan y x
z 1 2e y e y sin x
z 5xy
2 z/x 2 2 z/y 2 0.
f y x, y 0
f x x, y 0
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
f x, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
F x, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
f x, y, z 3x 2 y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
fx, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
f x, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
F x, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
f x, y, z 3x 2 y 5xyz 10yz 2
Hx, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
fx, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
Fx, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
fx, y, z 3x 2 y 5xyz 10yz 2
Hx, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
fx, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
f x, y ln x 2 y 2 1
f x, y e x2 xy y 2
f x, y 3x 3 12xy y 3
f x, y
1
x
1
y
xy
f x, y x 2 xy y 2
f x, y x 2 4xy y 2 4x 16y 3
f x, y x 2 xy y 2 5x y
f x, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
Fx, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
fx, y, z 3x 2 y 5xyz 10yz 2
Hx, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
f x, y, z
xyz
f yyx
f yxy ,
f xyy ,
f x, y
xy
x
y
f x, y
ln
x
x 2 y 2 f x, y 25 x 2 y 2
f x, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
F x, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
f x, y, z 3x 2 y 5xyz 10yz 2
Hx, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
f x, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
Fx, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
fx, y, z 3x 2 y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
fx, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
Fx, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
fx, y, z 3x 2 y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
f x, y, z tan y 2 z e z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
fx, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
Fx, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
fx, y, z 3x 2 y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
Desarrollo de conceptos
111. Sea f una función de dos variables x y y. Describir el procedimiento
para hallar las derivadas parciales de primer orden.
112. Dibujar una superficie que represente una función f de dos
variables x y y. Utilizar la gráfica para dar una interpretación
geométrica de
y
113. Dibujar la gráfica de una función cuya derivada
sea siempre negativa y cuya derivada
sea siempre positiva.
114. Dibujar la gráfica de una función cuyas derivadas
y sean siempre positivas.
115. Si es una función de y tal que y son continuas,
¿qué relación existe entre las derivadas parciales mixtas?
Explicar.
f yx
f xy
y
x
f
f y
f x
z fx, y
f y
f x
z fx, y
fy.
fx
In Exercises 59–64, find the first partial derivatives with respect
to
and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.
Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and such
that and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find the
first and second partial derivatives of the function. Determine
whether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives
and
are equal.
93.
94.
95.
96.
Laplace’s Equation
In Exercises 97–100, show that the function
satisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation
In Exercises 101–104, show that the function
satisfies the wave equation
101. 102.
103. 104.
Heat Equation
In Exercises 105 and 106, show that the function
satisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a
function
with the given partial derivatives. Explain your
reasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative with
respect to
109.
110. f x, y, z x senh y z
y 2 2 y 1 z
fx, y, z tan y 2 ze z2 y 2 z
x.
f y x, y x 4y
f x x, y 2x y,
f y x, y 2 sen 3x 2y
f x x, y 3 sen 3x 2y ,
f x, y
z
e t sen x c
z
e t cos x c
z/ t c 2 2 z/ x 2 .
z
sen ct sen x
z ln x ct
z cos 4x 4ct
z sen x ct
2 z/ t 2 c 2 2 z/ x 2 .
z
arctan y x
z
e x sen y
z
1
2 ey e y sen x
z
5xy
2 z/ x 2 2 z/ y 2 0.
f x, y, z
2z
x
y
f x, y, z
e x sen yz
f x, y, z x 2 3xy 4yz z 3
f x, y, z
xyz
f yyx
f yxy ,
f xyy ,
fx, y
xy
x
y
fx, y
ln
x
x 2 y 2 fx, y 25 x 2 y 2
fx, y
x sec y
f y x, y 0
f x x, y 0
y
x
fx, y ln x 2 y 2 1
fx, y e x2 xy y 2
fx, y 3x 3 12xy y 3
fx, y
1
x
1
y
xy
fx, y x 2 xy y 2
fx, y x 2 4xy y 2 4x 16y 3
fx, y x 2 xy y 2 5x y
fx, y x 2 xy y 2 2x 2y
f y x, y 0
f x x, y 0
y
x
f x, y ,
z
arctan y x
z
cos xy
z 2xe y 3ye x
z
e x tan y
z ln x y
z x 2 y 2 z x 4 3x 2 y 2 y 4
z x 2 2xy 3y 2 z x 2 3y 2
z 3xy 2 1, 2, 1
f x, y, z 3x 2 y 2 2z 2 ,
0, 2 , 4
f x, y, z z sen x y ,
3, 1, 1
f x, y, z
xy
x y z ,
fx, y, z
x
yz , 1, 1, 1 2, 1, 2
f x, y, z x 2 y 3 2xyz 3yz,
f x, y, z x 3 yz 2 , 1, 1, 1
f z
f y ,
f x ,
G x, y, z
1
1 x 2 y 2 z 2
F x, y, z ln x 2 y 2 z 2 w
7xz
x
y
w x 2 y 2 z 2
f x, y, z 3x 2 y 5xyz 10yz 2
Hx, y, z sen x 2y 3z
z.
y,
x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe the
procedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variables
and
Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivative
is always negative and whose derivative
is always
positive.
114. Sketch the graph of a function whose derivatives
and are always positive.
115. If is a function of and such that and are
continuous, what is the relationship between the mixed
partial derivatives? Explain.
f yx
f xy
y
x
f
f y
f x z f x, y
f y
f x z f x, y
f y.
f
x
y.
x
f
y.
x
f
WRITING ABOUT CONCEPTS
sen
sen