Bài tập toán cao cấp part 8 doc

9.1. D
-
a
.
o h`am riˆeng 111
2. Tu
.
o
.
ng tu
.
.
:nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
n
lim
∆y→0
∆
y
w
∆y
= lim
∆y→0

f(x, y +∆y) −f(x, y)
∆y
th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
il`ad
a
.
o h`am riˆeng cu
’
a h`am f(x, y) theo biˆe
´
n
y ta
.
idiˆe
’
m M(x, y)v`adu
.
o
.

.
cchı
’
bo
.
’
imˆo
.
t trong c´ac k´yhiˆe
.
u
∂w
∂y
,
∂f(x, y)
∂y
,f

y
(x, y),w

y
.
T`u
.
di
.
nh ngh˜ıa suy r˘a
`
ng da

.
o h`am riˆeng cu
’
a h`am hai biˆe
´
n theo biˆe
´
n
x l`a da
.
o h`am thˆong thu
.
`o
.
ng cu
’
a h`am mˆo
.
tbiˆe
´
n x khi cˆo
´
di
.
nh gi´a tri
.
cu
’
abiˆe
´

n y.Dod
´o c ´a c da
.
o h`am riˆeng du
.
o
.
.
c t´ınh theo c´ac quy t˘a
´
cv`a
cˆong th´u
.
c t´ınh da
.
o h`am thˆong thu
.
`o
.
ng cu
’
a h`am mˆo
.
tbiˆe
´
n.
Nhˆa
.
nx´et. Ho`an to`an tu
.

o
.
ng tu
.
.
ta c´o thˆe
’
di
.
nh ngh˜ıa da
.
o h`am riˆeng
cu
’
a h`am ba (ho˘a
.
c nhiˆe
`
uho
.
n ba) biˆe
´
nsˆo
´
.
9.1.2 D
-
a
.
o h`am cu

’
a h`am ho
.
.
p
Nˆe
´
u h`am w = f(x, y), x = x(t), y = y(t)th`ıbiˆe
’
uth´u
.
c w =
f[x(t),y(t)] l`a h`am ho
.
.
pcu
’
a t. Khi d
´o
dw
dt
=
∂w
∂x
·
dx
dt
+
∂w
∂y

·
dy
dt
· (9.1)
Nˆe
´
u w = f(x, y), trong d
´o x = x(u, v), y = y(u, v)th`ı







∂w
∂u
=
∂w
∂x
∂x
∂u
+
∂w
∂y
∂y
∂u
,
∂w
∂v

=
∂w
∂x
∂x
∂v
+
∂w
∂y
∂y
∂v
·
(9.2)
9.1.3 H`am kha
’
vi
Gia
’
su
.
’
h`am w = f(M) x´ac d
i
.
nh trong mˆo
.
t lˆan cˆa
.
n n`ao d´ocu
’
adiˆe

’
m
M(x, y). H`am f du
.
o
.
.
cgo
.
i l`a h`am kha
’
vi ta
.
idiˆe
’
m M(x, y)nˆe
´
usˆo
´
gia
112 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
∆f(M)=f(x +∆,y+∆y) − f(x, y)cu

’
a h`am khi chuyˆe
’
nt`u
.
d
iˆe
’
m
M(x, y)dˆe
´
ndiˆe
’
N(x +∆,y+∆y) c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.
´o
.
ida
.
ng
∆f(M)=D
1
∆x + D
2

∆y + o(ρ),ρ→ 0
trong d
´o ρ =

∆x
2
+∆y
2
.
Nˆe
´
u h`am f( x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y)th`ı
∂f
∂x
(M)=D
1
,
∂f
∂y
(M)=D
2
v`a khi d´o
∆f(M)=

∂f
∂x
(M)∆x +
∂f
∂y
∆y + o(ρ),ρ→ 0. (9.3)
9.1.4 D
-
a
.
o h`am theo hu
.
´o
.
ng
Gia
’
su
.
’
:
(1) w = f(M) l`a h`am x´ac di
.
nh trong lˆan cˆa
.
n n`ao d´o c u
’
adiˆe
’
m

M(x, y);
(2) e = (cos α, cos β) l`a vecto
.
do
.
nvi
.
trˆen du
.
`o
.
ng th˘a
’
ng c´o hu
.
´o
.
ng
L qua diˆe
’
m M(x, y);
(3) N = N(x +∆x, y +∆y)l`adiˆe
’
m thuˆo
.
c L v`a ∆e l`a dˆo
.
d`ai cu
’
a

doa
.
n th˘a
’
ng MN.
Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n
lim
∆→0
(N→M)
∆w
∆
th`ı gi´o
.
iha
.
nd´odu

.
o
.
.
cgo
.
il`ada
.
o h`am ta
.
idiˆe
’
m M(x, y) theo hu
.
´o
.
ng cu
’
a
vecto
.
e v`a du
.
o
.
.
ck´yhiˆe
.
ul`a
∂w

∂e
,t´u
.
cl`a
∂w
∂e
= lim
∆→0
∆w
∆
·
9.1. D
-
a
.
o h`am riˆeng 113
Da
.
o h`am theo hu
.
´o
.
ng cu
’
a vecto
.
e = (cos α,cos β)d
u
.
o

.
.
c t´ınh theo
cˆong th´u
.
c
∂f
∂e
=
∂f
∂x
(M) cos α +
∂f
∂y
(M) cos β. (9.4)
trong d
´o cos α v`a cos β l`a c´ac cosin chı
’
phu
.
o
.
ng cu
’
a vecto
.
e .
Vecto
.
v´o

.
i c´ac to
.
ad
ˆo
.
∂f
∂x
v`a
∂F
∂y
(t ´u
.
c l`a vecto
.

∂f
∂x
,
∂f
∂y

)d
u
.
o
.
.
cgo
.

i
l`a vecto
.
gradiˆen cu
’
a h`am f(M)ta
.
id
iˆe
’
m M(x, y)v`adu
.
o
.
.
ck´yhiˆe
.
ul`a
gradf(M).
T`u
.
d´o d a
.
o h`am theo hu
.
´o
.
ng
∂f
∂e

c´o biˆe
’
uth´u
.
cl`a
∂f
∂e
=

gradf,e

.
Ta lu
.
u´yr˘a
`
ng: 1) Nˆe
´
u h`am w = f(x, y) kha
’
vi ta
.
idiˆe
’
m M(x, y)
th`ı n´o liˆen tu
.
cta
.
i M v`a c´o c´ac da

.
o h`am riˆeng cˆa
´
p1ta
.
id´o ;
2) N´eu h`am w = f(x, y) c´o c´ac d
a
.
o h`am riˆeng cˆa
´
p 1 theo mo
.
ibiˆe
´
n
trong lˆan cˆa
.
nn`aod´ocu
’
adiˆe
’
m M(x, y) v`a c´ac da
.
o h`am riˆeng n`ay liˆen
tu
.
cta
.
idiˆe

’
m M(x, y) th`ı n´o kha
’
vi ta
.
idiˆe
’
m M.
Nˆe
´
u h`am f( x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y) th`ı n´o c´o da
.
o h`am theo
mo
.
ihu
.
´o
.
ng ta
.
idiˆe
’

md´o .
Ch´u´y.Nˆe
´
u h`am f(x, y)c´oda
.
o h`am theo mo
.
ihu
.
´o
.
ng ta
.
idiˆe
’
m M
0
th`ı khˆong c´o g`ıda
’
mba
’
o l`a h`am f(x, y) kha
’
vi ta
.
idiˆe
’
m M
0
(xem v´ı

du
.
4).
9.1.5 D
-
a
.
o h`am riˆeng cˆa
´
p cao
Gia
’
su
.
’
miˆe
`
n D ⊂ R
2
v`a
f : D → R
114 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n

l`a h`am hai biˆe
´
n f(x, y)du
.
o
.
.
c cho trˆen D.Tad
˘a
.
t
D
x
=

(x, y) ∈ D : ∃
∂f
∂x
= ±∞

,
D
y
=

(x, y) ∈ D : ∃
∂f
∂y
= ±∞


.
D
∗
= D
x
∩ D
y
D
-
i
.
nh ngh˜ıa. 1) C´ac da
.
o h`am riˆeng
∂f
∂x
v`a
∂f
∂y
du
.
o
.
.
cgo
.
i l`a c´ac d
a
.
o

h`am riˆeng cˆa
´
p1.
2) Nˆe
´
u h`am
∂f
∂x
: D
x
→ R v`a
∂f
∂y
: D
y
→ R c´o c´ac da
.
o h`am riˆeng
∂
∂x

∂f
∂x

=
∂
2
f
∂x∂x
=

∂
2
f
∂x
2
,
∂
∂y

∂f
∂x

=
∂
2
f
∂x∂y
,
∂
∂x

∂f
∂y

=
∂
2
f
∂y∂x
,

∂
∂y

∂f
∂y

=
∂
2
f
∂y∂y
=
∂
2
f
∂y
2
th`ı ch´ung du
.
o
.
.
cgo
.
i l`a c´ac da
.
o h`am riˆeng cˆa
´
p2theo x v`a theo y.
C´ac da

.
o h`am riˆeng cˆa
´
p3du
.
o
.
.
cdi
.
nh ngh˜ıa nhu
.
l`a c´ac da
.
o h`am riˆeng
cu
’
ada
.
o h`am riˆeng cˆa
´
p 2, v.v
Ta lu
.
u´yr˘a
`
ng nˆe
´
u h`am f(x, y) c´o c´ac da
.

o h`am hˆo
˜
nho
.
.
p
∂
2
f
∂x∂y
v`a
∂
2
f
∂y∂x
liˆen tu
.
cta
.
idiˆe
’
m(x,y) th`ı ta
.
idiˆe
’
md´o c´ac da
.
o h`am hˆo
˜
nho

.
.
p n`ay
b˘a
`
ng nhau:
∂
2
f
∂x∂y
=
∂
2
f
∂y∂x
·
C
´
AC V
´
IDU
.
9.1. D
-
a
.
o h`am riˆeng 115
V´ı du
.
1. T´ınh da

.
o h`am riˆeng cˆa
´
p1cu
’
a c´ac h`am
1) 4w = x
2
− 2xy
2
+ y
3
.2)w = x
y
.
Gia
’
i. 1) D
a
.
o h`am riˆeng
∂w
∂x
du
.
o
.
.
c t´ınh nhu
.

l`a d
a
.
o h`am cu
’
a h`am w
theo biˆe
´
n x v´o
.
i gia
’
thiˆe
´
t y = const. Do d´o
∂w
∂x
=(x
2
− 2xy
2
+ y
3
)

x
=2x − 2y
2
+0=2(x −y
2

).
Tu
.
o
.
ng tu
.
.
, ta c´o
∂w
∂y
=(x
2
−2xy
2
+ y
3
)

y
=0−4xy +3y
2
= y(3y − 4x).
2) Nhu
.
trong 1), xem y = const ta c´o
∂w
∂x
=


x
y


x
= yx
y−1
.
Tu
.
o
.
ng tu
.
.
, khi xem x l`a h˘a
`
ng sˆo
´
ta thu d
u
.
o
.
.
c
∂w
∂y
= x
y

lnx.
(v`ı w = x
y
l`a h`am m˜udˆo
´
iv´o
.
ibiˆe
´
n y khi x = const. 
V´ı d u
.
2. Cho w = f(x, y)v`ax = ρ cos ϕ, y = ρ sin ϕ. H˜ay t´ınh
∂w
∂ρ
v`a
∂w
∂ϕ
.
Gia
’
i. D
ˆe
’
´ap du
.
ng cˆong th´u
.
c (9.2), ta lu
.

u´yr˘a
`
ng
w = f( x, y)=f(ρ cos ϕ, ρ sin ϕ)=F (ρ, ϕ).
Do d´o theo (9.2) v`a biˆe
’
uth´u
.
cdˆo
´
iv´o
.
i x v`a y ta c´o
∂w
∂ρ
=
∂w
∂x
∂x
∂ρ
+
∂w
∂y
∂y
∂ρ
=
∂w
∂x
cos ϕ +
∂w

∂y
sin ϕ
∂w
∂ϕ
=
∂w
∂x
∂x
∂ϕ
+
∂w
∂y
∂y
∂ϕ
=
∂w
∂x
(−ρ sin ϕ)+
∂w
∂y
(ρ cos ϕ)
= ρ

−
∂w
∂x
sin ϕ +
∂w
∂y
cos ϕ


. 
116 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
V´ı du
.
3. T´ınh da
.
o h`am cu
’
a h`am w = x
2
+ y
2
x ta
.
idiˆe
’
m M
0
(1, 2) theo
hu
.

´o
.
ng cu
’
a vecto
.
−→
M
0
M
1
, trong d´o M
1
l`a diˆe
’
mv´o
.
ito
.
ad
ˆo
.
(3, 0).
Gia
’
i. D
ˆa
`
u tiˆen ta t`ım vecto
.

d
o
.
nvi
.
e c´o hu
.
´o
.
ng l`a hu
.
´o
.
ng d
˜a cho.
Ta c´o
−→
M
0
M
1
=(2,−2)=2e
1
− 2e
2
,
⇒|
−→
M
0

M
1
| =2
√
2 ⇒ e =
M
0
M
1
|M
0
M
1
|
=
2e
1
− 2e
2
2
√
2
=
1
√
2
e
1
−
1

√
2
e
2
.
trong d
´o e
1
, e
2
l`a vecto
.
do
.
nvi
.
cu
’
a c´ac tru
.
cto
.
adˆo
.
.T`u
.
d´o suy r˘a
`
ng
cos α =

1
√
2
, cos β = −
1
√
2
·
Tiˆe
´
p theo ta t´ınh c´ac d
a
.
o h`am riˆeng ta
.
idiˆe
’
m M
0
(1, 2). Ta c´o
f

x
=2x + y
2
⇒ f

x
(M
0

)=f

x
(1, 2)=6,
f

y
=2xy ⇒ f

y
(M
0
)=f

y
(1, 2)=4.
Do d
´o theo cˆong th´u
.
c (9.4) ta thu d
u
.
o
.
.
c
∂f
∂e
=6·
1

√
2
− 4 ·
1
√
2
=
√
2. 
V´ı d u
.
4. H`am f( x, y)=x + y +

|xy| c´o da
.
o h`am theo mo
.
ihu
.
´o
.
ng
ta
.
idiˆe
’
m O(0, 0) nhu
.
ng khˆong kha
’

vi ta
.
id´o.
Gia
’
i. 1. Su
.
.
tˆo
`
nta
.
id
a
.
o h`am theo mo
.
ihu
.
´o
.
ng.
Ta x´et hu
.
´o
.
ng cu
’
a vecto
.

e d
irat`u
.
Ov`alˆa
.
pv´o
.
i tru
.
c Ox g´oc α.Ta
c´o
∆
e
f(0, 0) = ∆x +∆y +

|∆x∆y|
=

cos α + sin α +

|cos α sin α|

ρ,
9.1. D
-
a
.
o h`am riˆeng 117
trong d´o ρ =


∆x
2
+∆y
2
,∆x = ρ cos α,∆y = ρ sin α.
T`u
.
d
´o suy ra
∂f
∂e
(0, 0) = lim
ρ→0
∆
e
f(0, 0)
ρ
= cos α + sin α +

|sin α cos α|
t´u
.
cl`ad
a
.
o h`am theo hu
.
´o
.
ng tˆo

`
nta
.
i theo mo
.
ihu
.
´o
.
ng.
2. Tuy nhiˆen h`am d
˜a cho khˆong kha
’
vi ta
.
i O. Thˆa
.
tvˆa
.
y, ta c´o
∆f(0, 0) = f(∆x, ∆y) −f(0 , 0)=∆x +∆y +

|∆x||∆y|−0.
V`ı f

x
=1v`af

y
= 1 (ta

.
i sao ? ) nˆen nˆe
´
u f kha
’
vi ta
.
i O(0, 0) th`ı
∆f(0, 0) = ∆x +∆y +

|∆x∆y| =1·∆x +1· ∆y + ε(ρ)ρ
ε(ρ) → 0(ρ → 0),ρ=

∆x
2
+∆y
2
hay l`a lu
.
u´y∆x = ρ cos α,∆y = ρ sin α ta c´o
ε(ρ)=

|cos α sinα|.
Vˆe
´
pha
’
id˘a
’
ng th´u

.
c n`ay khˆong pha
’
i l`a vˆo c`ung b´e khi ρ → 0 (v`ı n´o
ho`an to`an khˆong phu
.
thuˆo
.
c v`ao ρ). Do d
´o theo di
.
nh ngh˜ıa h`am f(x, y)
d˜a cho khˆong kha
’
vi ta
.
idiˆe
’
mO.
V´ı du
.
5. T´ınh c´ac d
a
.
o h`am riˆeng cˆa
´
p2cu
’
a c´ac h`am:
1) w = x

y
,2)w = arctg
x
y
·
Gia
’
i. 1) D
ˆa
`
u tiˆen t´ınh c´ac da
.
o h`am riˆeng cˆa
´
p1.Tac´o
∂w
∂x
= yx
y−1
,
∂w
∂y
= x
y
lnx.
Tiˆe
´
p theo ta c´o
∂
2

w
∂x
2
= y(y −1)x
y−2
,
∂
2
w
∂y∂x
= x
y−1
+ yx
y−1
lnx = x
y−1
(1 + ylnx),
∂
2
w
∂x∂y
= yx
y−1
lnx + x
y
·
1
x
= x
y−1

(1 + ylnx),
∂
2
f
∂y
2
= x
y
(lnx)
2
.
118 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
2) Ta c´o
∂w
∂x
=
y
x
2
+ y
2
,

∂w
∂y
= −
x
x
2
+ y
2
·
T`u
.
d
´o
∂
2
w
∂x
2
=
∂
∂x

y
x
2
+ y
2

= −
2xy

(x
2
+ y
2
)
2
,
∂
2
w
∂y
2
=
∂
∂y

−x
x
2
+ y
2

=
2xy
x
2
+ y
2
,
∂

2
w
∂x∂y
=
∂
∂y

y
x
2
+ y
2

=
x
2
− y
2
(x
2
+ y
2
)
2
,
∂
2
w
∂y∂x
=

∂
∂x

−
x
x
2
+ y
2

=
x
2
−y
2
(x
2
+ y
2
)
2
·
Nhˆa
.
nx´et. Trong ca
’
1) lˆa
˜
n2)tadˆe
`

uc´o
∂
2
w
∂x∂y
=
∂
2
w
∂y∂x
. 
V´ı d u
.
6. T´ınh c´ac da
.
o h`am riˆeng cˆa
´
p1cu
’
a h`am w = f(x+y
2
,y+x
2
)
ta
.
id
iˆe
’
m M

0
(−1, 1), trong d´o x v`a y l`a biˆe
´
ndˆo
.
clˆa
.
p.
Gia
’
i. D
˘a
.
t t = x + y
2
, v = y + x
2
. Khi d´o
w = f( x + y
2
,y+ x
2
)=f(t, v).
Nhu
.
vˆa
.
y w = f( t, v) l`a h`am ho
.
.

pcu
’
a hai biˆe
´
ndˆo
.
clˆa
.
p x v`a y. N´o phu
.
thuˆo
.
c c´ac biˆe
´
ndˆo
.
clˆa
.
p thˆong qua hai biˆe
´
n trung gian t, v. Theo cˆong
th ´u
.
c (9.2) ta c´o:
∂w
∂x
=
∂f
∂t
·

∂t
∂x
+
∂f
∂v
·
∂v
∂x
= f

t
(x + y
2
,y+ x
2
) ·1+f

v
(x + y
2
,y+ x
2
) ·2x
= f

t
+2xf

v
.

9.1. D
-
a
.
o h`am riˆeng 119
∂w
∂x
(−1, 1) =
∂f
∂x
(0, 2) = f

t
(0, 2) −2f

v
(0, 2)
∂w
∂y
=
∂f
∂t
·
∂t
∂y
+
∂f
∂v
·
∂v

∂y
= f

t
(·)2y + f

v
(·)1
=2yf

t
+ f

v
∂w
∂y
(−1, 1) =
∂f
∂y
(0, 2)=2f

t
(0, 2) + f

v
(0, 2). 
B
`
AI T
ˆ

A
.
P
T´ınh d
a
.
o h`am riˆeng cu
’
a c´ac h`am sau dˆay
1. f(x, y)=x
2
+ y
3
+3x
2
y
3
.
(DS. f

x
=2x +6xy
3
, f

y
=3y
2
+9x
2

y
2
)
2. f(x, y, z)=xyz +
x
yz
.
(DS. f

x
= yz +
1
yz
, f

y
= xz −
x
y
2
z
, f

z
= xy −
x
yz
2
)
3. f(x, y, z) = sin(xy + yz). (D

S. f

x
= y cos(xy + yz),
f

y
=(x + z) cos(xy + yz), f

z
= y cos(xy + yz))
4. f(x, y) = tg(x + y)e
x/y
.
(D
S. f

x
=
e
x/y
cos
2
(x + y)
+ tg(x + y)e
x/y
1
y
,
f


y
=
e
x/y
cos
2
(x + y)
+ tg(x + y)e
x/y

−
x
y
2

.)
5. f = arc sin
x

x
2
+ y
2
.(DS. f

x
=
|y|
x

2
+ y
2
, f

y
=
−xsigny
x
2
+ y
2
)
6. f(x, y)=xyln(xy). (DS. f

x
= yln(xy)+y, f

y
= xln(xy)+x)
120 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
7. f( x, y, z)=


y
x

z
.
(DS. f

x
= z

y
x

z−1

−
y
x
2

= −
z
x

y
x

z
,

f

y
=
z
y

y
x

z
,f

z
=

y
x

z
ln
y
x
)
8. f( x, y, z)=z
x/y
.
(D
S. f


x
= x
x/y
lnz ·

1
y

, f

y
= z
x/y
lnz ·

−x
y
2

, f

z
=

x
y

z
x/y−1
)

9. f( x, y, z)=x
y
z
.
(DS. f

x
= y
z
x
y
z
−1
, f

y
= x
y
z
zy
z−1
lnx, f

z
= x
y
z
ln(x)
z
lny)

10. f( x, y, z)=x
y
y
z
z
x
.
(DS. f

x
= x
y−1
y
z+1
z
x
+ x
y
y
z
z
x
lnz, f

y
= x
y
lnxy
z
z

x
+ x
y
y
z−1
z
x+1
,
f

z
= x
y
y
z
lny · z
x
+ x
y+1
y
z
z
x−1
)
11. f( x, y) = ln sin
x + a
√
y
.
(D

S. f

x
=
1
√
y
cotg
x + a
√
y
, f

y
= −
x + a
y
cotg
x + a
√
y
)
12. f( x, y)=
x
y
− e
x
arctgy.
(DS. f


x
=
1
y
− e
x
arctgy, f

y
= −
x
y
2
−
e
x
1+y
2
)
13. f( x, y)=ln

x +

x
2
+ y
2

.
(D

S. f

x
=
1

x
2
+ y
2
, f

y
=
1
x +

x
2
+ y
2
·
y

x
2
+ y
2
).
T`ım da

.
o h`am riˆeng cu
’
a h`am ho
.
.
p sau d
ˆay (gia
’
thiˆe
´
t h`am f(x, y)
kha
’
vi)
14. f( x, y)=f(x + y, x
2
+ y
2
).
(DS. f

x
= f

t
+ f

v
2x, f


y
= f

t
+ f

v
2y, t = x + y, v = x
2
+ y
2
)
15. f( x, y)=f

x
y
,
y
x

.
9.1. D
-
a
.
o h`am riˆeng 121
(DS. f

x

=
1
y
f

t
−
y
x
2
f

v
, f

y
=
−x
y
2
f

t
+
1
x
f

v
, t =

x
y
, v =
y
x
)
16. f(x, y)=f( x − y,xy).
(DS. f

x
= f

t
+ yf

v
, f

y
= −f

t
+ xf

v
, t = x −y, v = xy)
17. f(x, y)=f( x − y
2
,y−x
2

,xy).
(DS. f

x
= f

t
− 2xf

v
+ yf

w
, f

y
= −2yf

t
+ f

v
+ xf

w
,
t = x − y
2
, v = y −x
2

, w = xy)
18. f(x, y, z)=f(

x
2
+ y
2
,

y
2
+ z
2
,
√
z
2
+ x
2
).
(DS. f

x
=
xf

t

x
2

+ y
2
+
xf

w
√
z
2
+ x
2
,f

y
=
yf

t

x
2
+ y
2
+
yf

v
√
x
2

+ z
2
,
f

z
=
zf

v

x
2
+ y
2
+
zf

w
√
z
2
+ x
2
,t=

x
2
+ y
2

,
v =

y
2
+ z
2
,w=
√
z
2
+ x
2
)
19. w = f(x, xy, xyz).
(D
S. f

x
= f

t
+ yf

u
+ yzf

v
,
f


y
= xf

u
+ xzf

v
,
f

z
= xyf

v
t = x, u = xy, v = xyz).
Trong c´ac b`ai to´an sau d
ˆay h˜ay ch´u
.
ng to
’
r˘a
`
ng h`am f(x, y) tho
’
a
m˜an phu
.
o
.

ng tr`ınh d˜a cho tu
.
o
.
ng ´u
.
ng (f(x, y)-kha
’
vi).
20. f = f(x
2
+ y
2
), y
∂f
∂x
− x
∂f
∂y
=0.
21. f = x
n
f

y
x
2

, x
∂f

∂y
+2y
∂f
∂y
= nf.
22. f = yf(x
2
− y
2
), y
2
∂f
∂x
+ xy
∂f
∂y
= xyf.
23. f =
y
2
3x
+ f(x, y), x
2
∂f
∂x
− xy
∂f
∂y
+ y
2

=0.
122 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
24. f = x
n
f

y
x
α
,
z
x
β

, x
∂f
∂x
+ αy
∂f
∂y
+ βz
∂f

∂z
= nf.
25. f =
xy
z
lnx + xf

y
x
,
z
x

, x
∂f
∂x
+ y
∂f
∂y
+ z
∂f
∂z
= f +
xy
z
.
26. T´ınh
∂
2
f

∂x
2
,
∂
2
f
∂x∂y
,
∂
2
f
∂y
2
nˆe
´
u f = cos(xy)
(D
S. f

xx
= −y
2
cos xy, f

xy
= −sin xy − xy cos xy, f

yy
=
−x

2
cos xy)
27. T´ınh c´ac da
.
o h`am riˆeng cˆa
´
p hai cu
’
a h`am f = sin(x + yz).
(DS. f

xx
= −sin t, f

xy
= −z sin t, f

xz
= −y sin t, f

yy
= −z
2
sin t,
f

yz
= −yz sin t, f

zz

= −y
2
sin t, t = x + yz)
28. T´ınh
∂
2
f
∂x∂y
nˆe
´
u f =

x
2
+ y
2
e
x+y
.
(DS.
e
x+y
(x
2
+ y
2
)
3/2

− xy +(x + y)(x

2
+ y
2
)+(x
2
+ y
2
)
2

)
29. T´ınh
∂
2
f
∂x∂y
,
∂
2
f
∂y∂z
,
∂
2
f
∂x∂z
nˆe
´
u f = x
yz

.
(D
S. f

xy
= x
yz−1
z(1 + yzlnx), f

xz
= x
yz−1
y(1 + yzlnx),
f

yz
=lnx · x
yz
(1 + yzlnx))
30. T´ınh
∂
2
f
∂x∂y
nˆe
´
u f = arctg
x + y
1 − xy
.(DS.

∂
2
f
∂x∂y
=0)
31. T´ınh f

xx
(0, 0), f

xy
(0, 0), f

yy
(0, 0) nˆe
´
u
f(x, y)=(1+x)
m
(1 + y)
n
.
(D
S. f

xx
(0, 0) = m(m − 1), f

xy
(0, 0) = mn, f


yy
(0, 0) = n(n −1))
32. T´ınh
∂
2
r
∂x
2
nˆe
´
u r =

x
2
+ y
2
+ z
2
.(DS.
r
2
− x
2
r
3
)
33. T´ınh f

xy

, f

yz
, f

xz
nˆe
´
u f =

x
y

z
.
(DS. f

xy
= −z
2
y
−2

xy
−1

z−1
, f

xz

=

1
y

x
y

z−1

1+zln
x
y

,
9.1. D
-
a
.
o h`am riˆeng 123
f

yz
= −
1
y

x
y


z
·

1+zln
x
y

)
34. Ch´u
.
ng minh r˘a
`
ng
∂
2
f
∂x∂y
=
∂
2
f
∂y∂x
nˆe
´
u f = arcsin

x −y
x
.
T´ınh c´ac d

a
.
o h`am cˆa
´
p hai cu
’
a c´ac h`am (gia
’
thiˆe
´
t hai lˆa
`
n kha
’
vi)
35. u = f(x + y,x
2
+ y
2
).
(D
S. u

xx
= f

tt
+4xf

tv

+4x
2
f

vv
+2f

v
,
u

xy
= f

tt
+2(x + y)f

tv
+4xyf

vv
,
u

yy
= f

tt
+4yf


tv
+4y
2
f

vv
+2f

v
,
t = x + y, v = x
2
+ y
2
.)
36. u = f

xy,
x
y

.
(D
S. u

xx
= y
2
f


tt
+2f

tv
+
1
y
2
f

vv
,
u

xy
= xyf

tt
−
x
y
3
f

vv
+ f

t
−
1

y
2
f

v
,
u

yy
= x
2
f

tt
−2
x
2
y
2
f

tv
+
x
2
y
4
f

vv

+
2x
y
3
f

v
,
t = xy, v =
x
y
)
37. u = f(sin x + cos y).
(D
S. u

xx
= cos
2
x ·f

−sin x ·f

, u

xy
= −sin y cos x ·f

,
u


yy
= sin
2
y · f

− cos y · f

)
38. Ch´u
.
ng minh r˘a
`
ng h`am
f =
1
2a
√
πt
e
−
(x−x
0
)
2
4a
2
t
(trong d´o a, x
0

l`a c´ac sˆo
´
) tho
’
a m˜an phu
.
o
.
ng tr`ınh truyˆe
`
n nhiˆe
.
t
∂f
∂t
= a
2
∂
2
f
∂x
2
·
124 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe

´
n
39. Ch´u
.
ng minh r˘a
`
ng h`am f =
1
r
trong d´o
r =

(x −x
0
)
2
+(y − y
0
)
2
+(z − z
0
)
2
tho
’
a m˜an phu
.
o
.

ng tr`ınh Laplace:
∆f ≡
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f
∂z
2
=0,r=0.
Trong c´ac b`ai to´an 40 - 44 ch´u
.
ng minh r˘a
`
ng c´ac h`am d˜a cho tho
’
a
m˜an phu
.
o

.
ng tr`ınh tu
.
o
.
ng ´u
.
ng (gia
’
thiˆe
´
t f v`a g l`a nh˜u
.
ng h`am hai lˆa
`
n
kha
’
vi)
40. u = f(x −at)+g(x + at),
∂
2
u
∂t
2
= a
2
∂
2
u

∂x
2
41. u = xf(x + y)+yg(x + y),
∂
2
u
∂x
2
−2
∂
2
u
∂x∂y
+
∂
2
u
∂y
2
=0.
42. u = f

y
x

+ xg

y
x


, x
2
∂
2
u
∂x
2
+2xy
∂
2
u
∂x∂y
+ y
2
∂
2
u
∂y
2
=0.
43. u = x
n
f

y
x

+ x
1−n
g


y
x

,
x
2
∂
2
u
∂x
2
+2xy
∂
2
u
∂x∂y
+ y
2
∂
2
u
∂y
2
= n(n −1)u.
44. u = f(x + g( y)),
∂u
∂x
·
∂

2
u
∂x∂y
=
∂u
∂y
·
∂
2
u
∂x
2
·
45. T`ım da
.
o h`am theo hu
.
´o
.
ng ϕ = 135
◦
cu
’
a h`am sˆo
´
f(x, y)=3x
4
+ xy + y
3
ta

.
idiˆe
’
m M(1, 2). (DS. −
√
2
2
)
46. T`ım d
a
.
o h`am cu
’
a h`am f(x, y)=x
3
− 3x
2
y +3xy
2
+1 ta
.
idiˆe
’
m
M(3, 1) theo hu
.
´o
.
ng t`u
.

diˆe
’
m n`ay dˆe
´
ndiˆe
’
m(6, 5). (DS. 0)
47. T`ım d
a
.
o h`am cu
’
a h`am f(x, y)=ln

x
2
+ y
2
ta
.
idiˆe
’
m M(1, 1)
theo hu
.
´o
.
ng phˆan gi´ac cu
’
a g´oc phˆa

`
ntu
.
th ´u
.
nhˆa
´
t. (D
S.
√
2
2
)
9.2. Vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
n 125
48. T`ım da
.
o h`am cu
’
a h`am f(x, y, z)=z
2
− 3xy +5 ta
.
idiˆe
’

m
M(1, 2, −1) theo hu
.
´o
.
ng lˆa
.
pv´o
.
i c´ac tru
.
cto
.
ad
ˆo
.
nh˜u
.
ng g´oc b˘a
`
ng nhau.
(D
S. −
√
3
3
)
49. T`ım da
.
o h`am cu

’
a h`am f( x, y, z)=ln(e
x
+ e
y
+ e
z
)ta
.
igˆo
´
cto
.
adˆo
.
v`a hu
.
´o
.
ng lˆa
.
pv´o
.
i c´ac tru
.
cto
.
adˆo
.
x, y, z c´ac g´oc tu

.
o
.
ng ´u
.
ng l`a α, β, γ.
(D
S.
cos α + cos β + cos γ
3
)
50. T´ınh da
.
o h`am cu
’
a h`am f(x, y)=2x
2
−3y
2
ta
.
idiˆe
’
m M(1, 0) theo
hu
.
´o
.
ng lˆa
.

pv´o
.
i tru
.
c ho`anh g´oc b˘a
`
ng 120
◦
.(DS. −2)
51. T`ım d
a
.
o h`am cu
’
a h`am z = x
2
−y
2
ta
.
idiˆe
’
m M
0
(1, 1) theo hu
.
´o
.
ng
vecto

.
e lˆa
.
pv´o
.
ihu
.
´o
.
ng du
.
o
.
ng tru
.
c ho`anh g´oc α =60
◦
.(DS. 1 −
√
3)
52. T`ım d
a
.
o h`am cu
’
a h`am z = ln(x
2
+ y
2
)ta

.
idiˆe
’
m M
0
(3, 4) theo
hu
.
´o
.
ng gradien cu
’
a h`am d´o. (DS.
2
5
)
53. T`ım gi´a tri
.
v`a hu
.
´o
.
ng cu
’
a vecto
.
gradien cu
’
a h`am
w =tgx −x + 3 sin y − sin

3
y + z + cotgz
ta
.
idiˆe
’
m M
0

π
4
,
π
3
,
π
2

.
(D
S. (gradw)
M
=

i +
3
8

j, cos α =
8

√
73
, cos β =
3
√
73
)
54. T`ım da
.
o h`am cu
’
a h`am w = arc sin
z

x
2
+ y
2
ta
.
idiˆe
’
m M
0
(1, 1, 1)
theo hu
.
´o
.
ng vecto

.
−→
M
0
M, trong d´o M =(3,2, 3). (DS.
1
6
)
9.2 Vi phˆan cu
’
ah`am nhiˆe
`
ubiˆe
´
n
Trong mu
.
c n`ay ta x´et vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
nm`adˆe
’
cho go
.
nta
chı
’

cˆa
`
n tr`ınh b`ay cho h`am hai biˆe
´
nl`ad
u
’
.Tru
.
`o
.
ng ho
.
.
psˆo
´
biˆe
´
nl´o
.
n
ho
.
nhaidu
.
o
.
.
c tr`ınh b`ay ho`an to`an tu
.

o
.
ng tu
.
.
.
126 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
9.2.1 Vi phˆan cˆa
´
p1
Gia
’
su
.
’
h`am w = f( x, y) kha
’
vi ta
.
idiˆe
’
m M(x, y), t´u

.
cl`ata
.
id´o s ˆo
´
gia
to`an phˆa
`
ncu
’
a h`am c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.
´o
.
ida
.
ng
∆f(M)=f(x +∆x, y +∆y) −f(x, y)
= D
1
∆x + D
2
∆y + o(ρ) (9.5)
trong d

´o ρ =

∆x
2
+∆y
2
, D
1
v`a D
2
khˆong phu
.
thuˆo
.
cv`ao∆x v`a
∆y. Khi d
´obiˆe
’
uth´u
.
c (go
.
il`aphˆa
`
nch´ınh tuyˆe
´
n t´ınh d
ˆo
´
iv´o

.
i ∆x v`a ∆y
cu
’
asˆo
´
gia ∆f)
D
1
∆x + D
2
∆y
d
u
.
o
.
.
cgo
.
il`avi phˆan (hay vi phˆan to`an phˆa
`
n ≡ hay vi phˆan th´u
.
nhˆa
´
t)
cu
’
a h`am w = f(x, y)v`adu

.
o
.
.
ck´yhiˆe
.
ul`adf :
df = D
1
∆x + D
2
∆y.
V`ı∆x = dx,∆y = dy v`a v`ı f(x, y) kha
’
vi ta
.
i M nˆen D
1
=
∂f
∂y
,
D
2
=
∂f
∂y
v`a
df =
∂f

∂x
dx +
∂f
∂y
dy (9.6)
Nhu
.
vˆa
.
y, nˆe
´
u w = f( x, y) kha
’
vi ta
.
i M(x, y)th`ıt`u
.
(9.5) v`a (9.6)
ta c´o
∆f(M)=df (M)+o(ρ)hay∆f(M)=df (M)+ε(ρ)ρ (9.7)
trong d
´o ε(ρ) → 0 khi ρ → 0.
9.2.2
´
Ap du
.
ng vi phˆan dˆe
’
t´ınh gˆa
`

nd´ung
Dˆo
´
iv´o
.
i∆x v`a ∆ y d
u
’
b´e ta c´o thˆe
’
thay xˆa
´
pxı
’
sˆo
´
gia ∆f(M)bo
.
’
ivi
phˆan df (M), t´u
.
cl`a
∆f(M) ≈ df (M)