Phép tính vi phân hàm một biến

Chu
.
o
.
ng 8
Ph´ep t´ınh vi phˆan h`am mˆo
.
t
biˆe
´
n
8.1 D
-
a
.
oh`am..................... 61
8.1.1 D
-
a
.
o h`am cˆa
´
p1................ 61
8.1.2 D
-
a
.
o h`am cˆa
´
pcao............... 62
8.2 Viphˆan ..................... 75

8.2.1 Vi phˆan cˆa
´
p1................. 75
8.2.2 Vi phˆan cˆa
´
pcao ............... 77
8.3 C´ac d
i
.
nh l´y co
.
ba
’
nvˆe
`
h`am kha
’
vi. Quy
t˘a
´
c l’Hospital. Cˆong th´u
.
c Taylor . . . . . . 84
8.3.1 C´ac d
i
.
nh l´y co
.
ba
’

nvˆe
`
h`am kha
’
vi ..... 84
8.3.2 Khu
.
’
c´ac da
.
ng vˆo d
i
.
nh. Quy t˘a
´
c Lˆopitan
(L’Hospitale) . . . . . . . . . . . . . . . . . 88
8.3.3 Cˆong th´u
.
cTaylor............... 96
8.1. D
-
a
.
o h`am 61
8.1 D
-
a
.
oh`am

8.1.1 D
-
a
.
o h`am cˆa
´
p1
Gia
’
su
.
’
h`am y = f(x)x´acd
i
.
nh trong δ-lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
(U(x
0
; δ)=
{x ∈ R : |x − x
0
| <δ)v`a∆f(x
0

)=f(x
0
+∆x) − f(x
0
) l`a sˆo
´
gia cu
’
a
n´o ta
.
id
iˆe
’
m x
0
tu
.
o
.
ng ´u
.
ng v´o
.
isˆo
´
gia ∆x = x − x
0
cu
’

adˆo
´
isˆo
´
.
Theo d
i
.
nh ngh˜ıa: Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n
lim
∆x→0
f(x
0
+∆x) − f(x
0
)

∆x
khi ∆x → 0 th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
il`ad
a
.
o h`am cu
’
a h`am f(x)ta
.
i
d
iˆe
’
m x
0
v`a du
.
o
.

.
cchı
’
bo
.
’
imˆo
.
t trong c´ac k´yhiˆe
.
u:
lim
∆x→0
f(x
0
+∆x) − f(x
0
)
∆x
≡
dy
dx
≡
d
dx
f(x) ≡ f

(x) ≡ y

.

D
a
.
ilu
.
o
.
.
ng
f

+
(x
0
)=f

(x
0
+ 0) = lim
∆x→0
∆x>0
∆y
∆x
= lim
∆x→0+0
∆y
∆x
v`a
f


−
(x
0
)=f

(x
0
− 0) = lim
∆x→0
∆x<0
∆y
∆x
= lim
∆x→0−0
∆y
∆x
d
u
.
o
.
.
cgo
.
il`ad
a
.
oh`ambˆen pha
’
i v`a da

.
oh`ambˆen tr´ai cu
’
a h`am y = f(x)
ta
.
id
iˆe
’
m x
0
nˆe
´
u c´ac gi´o
.
iha
.
nd
˜a n ˆe u t ˆo
`
nta
.
i.
Su
.
’
du
.
ng kh´ai niˆe
.

m gi´o
.
iha
.
nmˆo
.
tph´ıa ta c´o:
D
-
i
.
nh l´y 8.1.1. H`am y = f(x) c´o d
a
.
o h`am ta
.
idiˆe
’
m x khi v`a chı
’
khi
c´ac d
a
.
o h`am mˆo
.
tph´ıa tˆo
`
nta
.

iv`ab˘a
`
ng nhau:
f

(x +0)=f

(x − 0) = f

(x).
H`am f(x) kha
’
vi nˆe
´
un´oc´od
a
.
o h`am f

(x)h˜u
.
uha
.
n. H`am f(x) kha
’
vi liˆen tu
.
c nˆe
´
ud

a
.
o h`am f

(x)tˆo
`
nta
.
i v`a liˆen tu
.
c. Nˆe
´
u h`am f(x) kha
’
vi th`ı n´o liˆen tu
.
c. D
iˆe
`
u kh˘a
’
ng di
.
nh ngu
.
o
.
.
cla
.

i l`a khˆong d
´ung.
62 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
8.1.2 D
-
a
.
o h`am cˆa
´
p cao
Da
.
o h`am f

(x)du
.
o
.
.
cgo
.
il`ad

a
.
o h`am cˆa
´
p1(hay da
.
o h`am bˆa
.
c nhˆa
´
t).
D
a
.
o h`am cu
’
a f

(x)du
.
o
.
.
cgo
.
il`ad
a
.
o h`am cˆa
´

p hai (hay da
.
o h`am th´u
.
hai)cu
’
a h`am f(x)v`ad
u
.
o
.
.
ck´yhiˆe
.
ul`ay

hay f

(x). Da
.
o h`am cu
’
a
f

(x)du
.
o
.
.

cgo
.
il`ad
a
.
o h`am cˆa
´
p3(hay da
.
o h`am th´u
.
ba)cu
’
a h`am f(x)
v`a d
u
.
o
.
.
ck´yhiˆe
.
u y

hay f

(x) (hay y
(3)
, f
(3)

(x) v.v...
Ta c´o ba
’
ng d
a
.
o h`am cu
’
a c´ac h`am so
.
cˆa
´
pco
.
ba
’
n
f(x) f

(x) f
(n)
(x)
x
a
ax
a−1
a(a− 1)(a− 2)···(a− n +1)x
a−n
,
x>0

e
x
e
x
e
x
a
x
a
x
lnaa
x
(lna)
n
lnx
1
x
(−1)
n−1
(n − 1)!
1
x
n
, x>0
log
a
x
1
xlna
(−1)

n−1
(n − 1)!
1
x
n
lna
, x>0
sin x cos x sin

x +
nπ
2

8.1. D
-
a
.
o h`am 63
f(x) f

(x) f
(n)
(x)
cos x − sin x cos

x +
nπ
2

tgx

1
cos
2
x
cotgx −
1
sin
2
x
arc sin x
1
√
1 − x
2
, |x| < 1
arccosx −
1
√
1 − x
2
, |x| < 1
arctgx
1
1+x
2
arccotgx −
1
1+x
2
Viˆe

.
c t´ınh da
.
o h`am du
.
o
.
.
cdu
.
.
a trˆen c´ac quy t˘a
´
c sau d
ˆay.
1
+
d
dx
[u + v]=
d
dx
u +
d
dx
v.
2
+
d
dx

(αu)=α
du
dx
, α ∈ R.
3
+
d
dx
(uv)=v
du
dx
+ u
dv
dx
.
4
+
d
dx

u
v

=
1
v
2

v
du

dx
− u
dv
dx

, v =0.
5
+
d
dx
f[u(x)] =
df
du
·
du
dx
(d
a
.
o h`am cu
’
a h`am ho
.
.
p).
6
+
Nˆe
´
u h`am y = y(x) c´o h`am ngu

.
o
.
.
c x = x(y)v`a
dy
dx
≡ y

x
=0th`ı
dx
dy
≡ x

y
=
1
y

x
·
64 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´

n
7
+
Nˆe
´
u h`am y = y(x)du
.
o
.
.
c cho du
.
´o
.
ida
.
ng ˆa
’
nbo
.
’
ihˆe
.
th´u
.
c kha
’
vi
F (x, y)=0v`aF


y
=0th`ı
dy
dx
= −
F

x
F

y
trong d´o F

x
v`a F

y
l`a da
.
o h`am theo biˆe
´
ntu
.
o
.
ng ´u
.
ng cu
’
a h`am F (x, y)

khi xem biˆe
´
n kia khˆong d
ˆo
’
i.
8
+
Nˆe
´
u h`am y = y(x)du
.
o
.
.
cchodu
.
´o
.
ida
.
ng tham sˆo
´
x = x(t),
y = y(t)(x

(t) = 0) th`ı
dy
dx
=

y

(t)
x

(t)
·
9
+
d
n
dx
n
(αu + βv)=α
d
n
u
dx
n
+ β
d
n
v
dx
n
;
d
n
dx
n

uv =
n

k=0
C
k
n
d
n−k
dx
n−k
u
d
k
dx
k
v (quy t˘a
´
c Leibniz).
Nhˆa
.
nx´et. 1) Khi t´ınh d
a
.
o h`am cu
’
amˆo
.
tbiˆe
’

uth´u
.
cd
˜a cho ta c´o thˆe
’
biˆe
´
nd
ˆo
’
iso
.
bˆo
.
biˆe
’
uth´u
.
cd
´o sao cho qu´a tr`ınh t´ınh da
.
oh`amdo
.
n gia
’
n
ho
.
n. Ch˘a
’

ng ha
.
nnˆe
´
ubiˆe
’
uth´u
.
cd
´o l`a logarit th`ı c´o thˆe
’
su
.
’
du
.
ng c´ac
t´ınh chˆa
´
tcu
’
a logarit d
ˆe
’
biˆe
´
ndˆo
’
i... rˆo
`

it´ınhda
.
o h`am. Trong nhiˆe
`
u
tru
.
`o
.
ng ho
.
.
p khi t´ınh d
a
.
o h`am ta nˆen lˆa
´
y logarit h`am d˜a cho rˆo
`
i´ap
du
.
ng cˆong th´u
.
cd
a
.
o h`am loga
d
dx

lny(x)=
y

(x)
y(x)
·
2) Nˆe
´
u h`am kha
’
vi trˆen mˆo
.
t khoa
’
ng d
u
.
o
.
.
cchobo
.
’
iphu
.
o
.
ng tr`ınh
F (x, y)=0th`ıd
a

.
o h`am y

(x) c´o thˆe
’
t`ım t `u
.
phu
.
o
.
ng tr`ınh
d
dx
F (x, y)=0.
C
´
AC V
´
IDU
.
8.1. D
-
a
.
o h`am 65
V´ı du
.
1. T´ınh da
.

o h`am y

nˆe
´
u:
1) y =ln
3

e
x
1 + cos x
; x = π(2n + 1), n ∈ N
2) y =
1+x
2
3
√
x
4
sin
7
x
, x = πn, n ∈ N.
Gia
’
i. 1) Tru
.
´o
.
chˆe

´
ttad
o
.
n gia
’
nbiˆe
’
uth´u
.
ccu
’
a h`am y b˘a
`
ng c´ach
du
.
.
a v`ao c´ac t´ınh chˆa
´
tcu
’
a logarit. Ta c´o
y =
1
3
lne
x
−
1

3
ln(1 + cos x)=
x
3
−
1
3
ln(1 + cos x).
Do d
´o
y

=
1
3
−
1
3
(cos x)

1 + cos x
=
1
3
+
1
3
sin x
1+cosx
=

1+tg
x
2
3
·
2) O
.
’
d
ˆay tiˆe
.
nlo
.
.
iho
.
nca
’
l`a x´et h`am z =ln|y|.Tac´o
dz
dx
=
dz
dy
·
dy
dx
=
1
y

dy
dx
⇒
dy
dx
= y
dz
dx
· (*)
Viˆe
´
t h`am z du
.
´o
.
ida
.
ng
x =ln|y| = ln(1 + x
2
) −
4
3
ln|x|−7ln| sin x|
⇒
dz
dx
=
2x
1+x

2
−
4
3x
− 7
cos x
sin x
·
Thˆe
´
biˆe
’
uth´u
.
cv`u
.
athud
u
.
o
.
.
cv`ao(∗) ta c´o
dy
dx
=
1+x
2
3
√

x
4
sin
7
x

2x
1+x
2
−
4
3x
− 7
cos x
sin x

. 
V´ı d u
.
2. T´ınh d
a
.
o h`am y

nˆe
´
u: 1) y = (2+cos x)
x
, x ∈ R;2)y = x
2

x
,
x>0.
Gia
’
i. 1) Theo d
i
.
nh ngh˜ıa ta c´o
y = e
xln(2+cosx)
.
66 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
T`u
.
d
´o
y

= e
xln(2+cosx)


xln(2 + cos x)


= e
xln(2+cosx)

ln(2 + cos x)− x
sin x
2 + cos x

,x∈ R.
2) V`ı y = e
2
x
lnx
nˆen v´o
.
i x>0 ta c´o
y

= e
2
x
lnx
[2
x
lnx]

= e
2

x
lnx

1
x
2
x
+2
x
ln2 · lnx

=2
x
x
2
x

1
x
+ln2· lnx

. 
V´ı du
.
3. T´ınh d
a
.
o h`am cˆa
´
p2cu

’
a h`am ngu
.
o
.
.
cv´o
.
i h`am y = x + x
5
,
x ∈ R.
Gia
’
i. H`am d
˜a cho liˆen tu
.
cv`ado
.
nd
iˆe
.
u kh˘a
´
pno
.
i, d
a
.
o h`am y


=
1+5x
4
khˆong triˆe
.
t tiˆeu ta
.
ibˆa
´
tc´u
.
d
iˆe
’
m n`ao. Do d´o
x

y
=
1
y

x
=
1
1+5x
4
·
Lˆa

´
yd
a
.
o h`am d˘a
’
ng th´u
.
c n`ay theo y ta thu d
u
.
o
.
.
c
x

yy
=

1
1+5x
4


x
· x

y
=

−20x
3
(1 + 5x
4
)
3
· 
V´ı d u
.
4. Gia
’
su
.
’
h`am y = f(x)d
u
.
o
.
.
cchodu
.
´o
.
ida
.
ng tham sˆo
´
bo
.

’
i c´ac
cˆong th´u
.
c x = x(t), y = y(t), t ∈ (a; b) v`a gia
’
su
.
’
x(t), y(t) kha
’
vi cˆa
´
p
2v`ax

(t) =0t ∈ (a, b). T`ım y

xx
.
Gia
’
i. Ta c´o
dy
dx
=
dy
dt
dx
dt

=
y

t
x

t
⇒ y

x
=
y

t
x

t
·
Lˆa
´
yd
a
.
o h`am hai vˆe
´
cu
’
ad˘a
’
ng th´u

.
c n`ay ta c´o
y

xx
=

y

t
x

t


t
· t

x
=

y

t
x

t


t

·
1
x

t
=
x

t
y

tt
− y

t
x

tt
x

t
3
· 
8.1. D
-
a
.
o h`am 67
V´ı du
.

5. Gia
’
su
.
’
y = y(x), |x| >al`a h`am gi´a tri
.
du
.
o
.
ng cho du
.
´o
.
i
da
.
ng ˆa
’
nbo
.
’
iphu
.
o
.
ng tr`ınh
x
2

a
2
−
y
2
b
2
=1.
T´ınh y

xx
.
Gia
’
i. D
ˆe
’
t`ım y

ta ´ap du
.
ng cˆong th´u
.
c
d
dx
F (x, y)=0.
Trong tru
.
`o

.
ng ho
.
.
p n`ay ta c´o
d
dx

x
2
a
2
−
y
2
b
2
− 1

=0.
Lˆa
´
yd
a
.
o h`am ta c´o
2x
a
2
−

2y
b
2
y

x
=0, (8.1)
⇒y

x
=
b
2
x
a
2
y
, |x| > 0,y >0. (8.2)
Lˆa
´
yd
a
.
o h`am (8.1) theo x ta thu du
.
o
.
.
c
1

a
2
−
1
b
2

y

x

2
−
y
b
2
y

xx
=0
v`a t`u
.
(8.2) ta thu d
u
.
o
.
.
c y


x
:
y

xx
=
1
y

b
2
a
2
−

y

x

2

=
1
y

b
2
a
2
−

b
4
a
4
x
2
y
2

= −
b
4
a
2
y
3

x
2
a
2
−
y
2
b
2

= −
b
4

a
2
y
3
,y>0. 
V´ı du
.
6. T´ınh y
(n)
nˆe
´
u: 1) y =
1
x
2
− 4
;2)y = x
2
cos 2x.
Gia
’
i. 1) Biˆe
’
udiˆe
˜
nh`amd
˜achodu
.
´o
.

ida
.
ng tˆo
’
ng c´ac phˆan th´u
.
cco
.
ba
’
n
1
x
2
− 4
=
1
4

1
x − 2
−
1
x +2

68 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo

.
tbiˆe
´
n
v`a khi d´o

1
x
2
− 4

(n)
=
1
4

1
x− 2

(n)
−

1
x +2

(n)

.
Do


1
x ± 2

(n)
=(−1)(−2)···(−1 − n + 1)(x ± 2)
−1−n
=(−1)
n
n!
1
(x± 2)
n+1
nˆen

1
x
2
− 4

(n)
=
(−1)
n
n!
4

1
(x − 2)
n+1
−

1
(x +2)
n+1

.
2) Ta ´ap du
.
ng cˆong th´u
.
c Leibniz d
ˆo
´
iv´o
.
id
a
.
o h`am cu
’
at´ıch
(x
2
cos 2x)=C
0
n
x
2
(cos 2x)
(n)
+ C

1
n
(x
2
)

(cos 2x)
n−1
+ C
2
n
(x
2
)

(cos 2x)
n−2
.
C´ac sˆo
´
ha
.
ng c`on la
.
id
ˆe
`
u=0v`ı

x

2

(k)
=0 ∀ k>2.
´
Ap du
.
ng cˆong th´u
.
c
(cos 2x)
(n)
=2
n
cos

2x +
nπ
2

ta thu d
u
.
o
.
.
c
(x
2
cos 2x)

(n)
=2
n

x
2
−
n(n − 1)
4

cos

2x +
nπ
2

+2
n
nx sin

2x +
nπ
2

. 
V´ı d u
.
7. V´o
.
i gi´a tri

.
n`ao cu
’
a a v`a b th`ı h`am
f(x)=



e
x
,x 0,
x
2
+ ax + b, x > 0
8.1. D
-
a
.
o h`am 69
c´o da
.
o h`am trˆen to`an tru
.
csˆo
´
.
Gia
’
i. R˜o r`ang l`a h`am f(x)c´od
a

.
o h`am ∀ x>0v`a∀ x<0. Ta chı
’
cˆa
`
nx´etd
iˆe
’
m x
0
=0.
V`ı h`am f(x) pha
’
i liˆen tu
.
cta
.
id
iˆe
’
m x
0
=0nˆen
lim
x→0+0
f(x) = lim
x→0−0
f(x) = lim
x→0
f(x)

t´u
.
cl`a
lim
x→0+0
(x
2
+ ax + b)=b = e
0
=1⇒ b =1.
Tiˆe
´
pd
´o, f

+
(0) = (x

+ ax + b)



x
0
=0
= a v`a f

−
(0) = e
x



x
0
=0
=1.
Do d
´o f

(0) tˆo
`
nta
.
inˆe
´
u a =1v`ab = 1. Nhu
.
vˆa
.
yv´o
.
i a =1,b =1
h`am d
˜a cho c´o da
.
o h`am ∀ x ∈ R. 
B
`
AI T
ˆ

A
.
P
T´ınh d
a
.
o h`am y

cu
’
a h`am y = f(x)nˆe
´
u:
1. y =
4
√
x
3
+
5
x
2
−
3
x
3
+ 2. (DS.
3
4
4

√
x
−
10
x
3
+
9
x
4
)
2. y = log
2
x + 3log
3
x.(DS.
ln24
xln2 · ln3
)
3. y =5
x
+6
x
+

1
7

x
.(DS. 5

x
ln5 + 6
x
ln6 − 7
−x
ln7)
4. y = ln(x +1+
√
x
2
+2x + 3). (DS.
1
√
x
2
+2x +3
)
5. y = tg5x.(D
S.
10
sin 10x
)
6. y = ln(ln
√
x). (DS.
1
2xln
√
x
)

7. y =ln

1+2x
1 − 2x
.(D
S.
2
1 − 4x
2
)
70 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
8. y = xarctg
√
2x − 1 −
√
2x − 1
2
.(D
S. arctg
√
2x − 1)
9. y = sin

2
x
3
.(DS. 3x
2
sin 2x
3
)
10. y = sin
4
x + cos
4
x.(DS. − sin 4x)
11. y =
√
xe
√
x
.(DS.
e
√
x
(1 +
√
x)
2
√
x
)
12. y = e

1
cos x
.(D
S. e
1
cos x
sin x
cos
2
x
)
13. y = e
1
lnx
.(D
S.
−e
1
lnx
xln
2
x
)
14. y =ln

e
2x
+
√
e

4x
+1. (DS.
2e
2x
√
e
4x
+1
)
15. y =ln

e
4x
e
4x
+1
.(D
S.
2
e
4x
+1
)
16. y = log
5
cos 7x.(DS. −
7tg7x
ln5
)
17. y = log

7
cos
√
1+x.(DS. −
tg
√
1+x
2
√
1+xln7
)
18. y = arccos

e
−
x
2
2

.(D
S.
xe
−
x
2
2
√
1 − e
−x
2

)
19. y = tg sin cos x.(D
S.
− sin cos(cos x)
cos
2
(sin cos x)
)
20. y = e
x
2
cotg3x
.(DS.
xe
c
2
cotg3x
sin
2
3x
(sin 6x − 3x))
21. y = e
√
1+lnx
.(DS.
e
√
1+lnx
2x
√

1+lnx
)
22. y = x
1
x
.(DS. x
1
x
−2
(1 − lnx))
23. y = e
x
.(DS. x
x
(1 + lnx))
8.1. D
-
a
.
o h`am 71
24. y = x
sin x
.(DS. x
sin x
cos x · lnx + x
sin x−1
sin x)
25. y = (tgx)
sin x
.(DS. (tgx)

sin x

cos xlntgx +
1
cos x

)
26. y = x
sin x
.(DS. x
sin x

sin x
x
+lnx· cos x

)
27. y = x
x
2
.(DS. x
x
2
+1
(1 + 2lnx))
28. y = x
e
x
.(DS. e
x

x
e
x

1
x
+lnx))
29. y = log
x
7. (DS. −
1
xlnxlog
7
x
)
30. y =
1
2a
ln



x − a
x + a



.(D
S.
1

x
2
− a
2
)
31. y = sin ln|x|.(D
S.
cos ln|x|
x
)
32. y =ln| sin x|.(D
S. cotgx)
33. y =ln|x +
√
x
2
+1|.(DS.
1
√
x
2
+1
).
Trong c´ac b`ai to´an sau d
ˆay (34-40) t´ınh da
.
o h`am cu
’
a h`am y du
.

o
.
.
c
cho du
.
´o
.
ida
.
ng tham sˆo
´
.
34. x = a cos t, a sin t, t ∈ (0,π). y

xx
?(DS. −
1
a sin
3
t
)
35. x = t
3
, y = t
2
. y

xx
?(DS. −

2
9t
4
)
36. x =1+e
at
, y = at + e
−at
. y

xx
?(DS. 2e
−3at
− e
−2at
)
37. x = a cos
3
t, y = a sin
3
t. y

xx
?(DS.
1
3a sin t cos
4
t
)
38. x = e

t
cos t, y = e
t
sin t. y

xx
?(DS.
2
e
t
(cos t − sin t)
3
)
39. x = t − sin t, y =1− cos t. y

xx
?(DS. −
1
4 sin
4
t
2
)
40. x = t
2
+2t, y = ln(1 + t). y

xx
?(DS.
−1

4(1 + t)
4
).
72 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
Trong c´ac b`ai to´an sau dˆay (41-47) t´ınh da
.
o h`am y

ho˘a
.
c y

cu
’
a
h`am ˆa
’
nd
u
.
o
.

.
c x´ac d
i
.
nh bo
.
’
i c´ac phu
.
o
.
ng tr`ınh d
˜acho
41. x +
√
xy + y = a. y

?(DS.
2a − 2x − y
x +2y − a
)
42. arctg
y
x
=ln

x
2
+ y
2

. y

?(DS.
x + y
x − y
)
43. e
x
sin y − e
−y
cos x =0. y

?(DS. −
e
x
sin y + e
−y
sin x
e
x
cos y + e
−y
cos x
)
44. x
2
y + arctg

y
x


=0. y

?(DS.
−2x
3
y − 2xy
3
+ y
x
4
+ x
2
y
2
+ x
)
45. e
x
− e
y
= y − x. y

?(DS.
(e
y
− e
x
)(e
x+y

− 1)
(e
y
+1)
3
)
46. x + y = e
x−y
. y

?(DS.
4(x + y)
(x + y +1)
3
)
47. y = x + arctgy. y

?(DS.
−(2y
2
+2)
y
5
).
Trong c´ac b`ai to´an sau d
ˆay (48-52) t´ınh da
.
o h`am cu
’
a h`am ngu

.
o
.
.
c
v´o
.
ih`amd
˜a cho.
48. y = x + x
3
, x ∈ R. x

y
?(DS. x

y
=
1
1+3x
2
)
49. y = x +lnx, x>0. x

y
?(DS. x

y
=
x

x +1
, y>0)
50. y = x + e
x
. x

y
?(DS. x

y
=
1
1+y − x
, y ∈ R)
51. y =chx, x>0. x

y
?(DS. x

y
=
1

y
2
− 1
)
52. y =
x
2

1+x
2
, x<0. x

y
?(DS. x

y
=
x
3
2y
2
, y ∈ (0, 1)).
53. V´o
.
i gi´a tri
.
n`ao cu
’
a a v`a b th`ı h`am
f(x)=



x
3
nˆe
´
u x  x

0
,
ax + b nˆe
´
u x>x
0
8.1. D
-
a
.
o h`am 73
liˆen tu
.
c v`a kha
’
vi ta
.
idiˆe
’
m x = x
0
?
(D
S. a =3x
2
0
, b = −2x
3
0
).

54. X´ac d
i
.
nh α v`a β dˆe
’
c´ac h`am sau: a) liˆen tu
.
c kh˘a
´
pno
.
i; b) kha
’
vi
kh˘a
´
pno
.
inˆe
´
u
1) f(x)=



αx + β nˆe
´
u x  1
x
2

nˆe
´
u x>1
2) f(x)=





α + βx
2
nˆe
´
u |x| < 1,
1
|x|
nˆe
´
u |x|  1.
(D
S. 1) a) α + β = 1, b) α =2,β = −1; 2) a) α + β = 1, b)
α =
3
2
,β = −
1
2
).
55. Gia
’

su
.
’
h`am y = f(x) x´ac d
i
.
nh trˆen tia (−∞,x
0
) v`a kha
’
vi bˆen
tr´ai ta
.
id
iˆe
’
m x = x
0
.V´o
.
i gi´a tri
.
n`ao cu
’
a a v`a b th`ı h`am
f(x)=



f(x)nˆe

´
u x  x
0
,
ax
2
+ b nˆe
´
u x>x
0
kha
’
vi ta
.
idiˆe
’
m x = x
0
(x
0
=0)?
(D
S. a =
f

(x
0
− 0)
2x
0

, b = f(x
0
) −
x
0
2
f

(x
0
− 0)).
Trong c´ac b`ai to´an (56-62) t´ınh d
a
.
o h`am y

nˆe
´
u
56. y = e
−x
2
.(DS. 2e
−x
2
(2x
2
− 1))
57. y =tgx.(D
S.

2 sin x
cos
3
x
)
58. y =
√
1+x
2
.(DS.
1
(1 + x
2
)
3/2
)
59. y = arcsin
x
2
.(D
S.
x
(4 − x
2
)
3/2
)
60. y = arctg
1
x

.(D
S.
2x
(1 + x
2
)
2
)
74 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
61. y = x arcsinx.(DS.
2 − x
2
(1 − x
2
)
√
1 − x
2
)
62. y = f(e
x
). (DS. e

x
f

(e
x
)+e
2x
f

(e
x
)).
Trong c´ac b`ai to´an (63-69) t´ınh d
a
.
o h`am cˆa
´
p3cu
’
a y nˆe
´
u:
63. y = arctg
x
2
.(D
S.
4(3x − 4)
(4 + x
2

)
3
)
64. y = xe
−x
.(DS. e
−x
(3 − x))
65. y = e
x
cos x.(DS. −2e
x
(cos x + sin x))
66. y = x
2
sin x.(DS. −2e
x
(cos x + sin x))
67. y = x
3
2
x
.(DS. 2
x
(x
3
ln
3
2+9x
2

ln
2
x +18xln2 + 6))
68. y = x
2
sin 2x.(DS. −4(2x
2
cos 2x +6x sin 2x − 3 cos 2x))
69. y =(f(x
2
). (DS. 12xf

(x
2
)+8x
3
f

(x
2
)).
Trong c´ac b`ai to´an (70-84) t´ınh d
a
.
o h`am y
(n)
nˆe
´
u
70. y = sin 3x.(D

S. 3
n
sin

3x +
nπ
2

)
72. y = e
x
2
.(DS. e
x
2

1
2

n
)
73. y =2
3x
.(DS. 2
3x
(3ln2)
n
)
74. y = cos
2

x.(DS. 2
n−1
cos

2x + n ·
π
2

)
75. y =(4x +1)
n
.(DS. 4
n
n!)
76. y =ln(ax + b). (D
S. (−1)
n−1
(n − 1)!
a
n
(ax + b)
n
)
77. y = sin
4
x + cos
4
x.(DS. 4
n−1
cos


4x +
nπ
2

)
Chı
’
dˆa
˜
n. Ch´u
.
ng minh r˘a
`
ng sin
4
x + cos
4
x =
3
4
+
1
4
cos 4x.
78. y = sin
3
x.(DS.
3
4

sin

x +
nπ
2

−
3
n
4
sin

3x + n ·
π
2

)
Chı
’
dˆa
˜
n. D`ung cˆong th´u
.
c sin 3x = 3 sin x − 4 sin
3
x.
8.2. Vi phˆan 75
79. y = sin αx sin βx.
(D
S.

1
2
(α− β)
n
cos[(α− β)x + n
π
2
]−
1
2
(α + β)
n
cos[(α + β)x + n
π
2
])
Chı
’
dˆa
˜
n. Biˆe
´
nd
ˆo
’
i t´ıch th`anh tˆo
’
ng.
80. y = e
αx

sin βx.
(D
S. e
αx

sin βx

α
n
−
n(n − 1)
1 · 2
α
n−2
β
2
+ ...

+
+ cos βx

nα
n−1
β −
n(n − 1)(n − 2)
1 · 2 · 3
α
n−3
β
3

+ ...

)
Chı
’
dˆa
˜
n. D`ung quy t˘a
´
c Leibniz.
81. y = e
x
(3x
2
− 4). (DS. e
x
[3x
2
+6nx +3n(n − 1) − 4])
82. y =ln
ax + b
ax− b

ax + b
ax− b
> 0

(D
S. (−1)
n−1

a
n
(n − 1)!

1
(ax + b)
n
−
1
ax − b)
n

)
83. y =
x
x
2
− 4x − 12
.(D
S.
(−1)
n
n!
4

3
(x − 6)
n+1
+
1

(x − 2)
n+1

)
84. y =
3 − 2x
2
2x
2
+3x − 2
.(D
S. (−1)
n
n!

2
n
(2x − 1)
n+1
+
1
(x +2)
n+1

)
Chı
’
dˆa
˜
n. D

ˆe
’
gia
’
i b`ai 83 v`a 84 cˆa
`
nbiˆe
’
udiˆe
˜
nh`amd˜a cho du
.
´o
.
ida
.
ng
tˆo
’
ng c´ac phˆan th´u
.
cd
o
.
n gia
’
n.
8.2 Vi phˆan
8.2.1 Vi phˆan cˆa
´

p1
Gia
’
su
.
’
h`am y = f(x) x´ac d
i
.
nh trong lˆan cˆa
.
n n`ao d´ocu
’
adiˆe
’
m x
0
v`a
∆x = x− x
0
l`a sˆo
´
gia cu
’
abiˆe
´
ndˆo
.
clˆa
.

p. H`am y = f(x)c´ovi phˆan cˆa
´
p
1 (vi phˆan th´u
.
nhˆa
´
t)ta
.
id
iˆe
’
m x
0
nˆe
´
u khi dˆo
´
isˆo
´
di
.
ch chuyˆe
’
nt`u
.
gi´a tri
.
x = x
0

dˆe
´
n gi´a tri
.
x = x
0
+∆x sˆo
´
gia tu
.
o
.
ng ´u
.
ng cu
’
a h`am f(x) c´o thˆe
’
76 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n
biˆe
’
udiˆe

˜
ndu
.
´o
.
ida
.
ng
∆f(x
0
) ≡ f(x
0
+∆x) − f(x
0
)=D(x
0
)∆x + o(∆x) (8.3)
trong d
´o D(x
0
) khˆong phu
.
thuˆo
.
c ∆x v`a
o(∆x)
∆x
→ 0 khi ∆x → 0. T´ıch
D(x
0

)∆x du
.
o
.
.
cgo
.
il`avi phˆan cˆa
´
p1cu
’
a h`am f(x)ta
.
id
iˆe
’
m x
0
v`a du
.
o
.
.
c
k´yhiˆe
.
u
dy ≡ df ≡
dy
dx

dx.
Sˆo
´
gia ∆x cu
’
abiˆe
´
nd
ˆo
.
clˆa
.
p x du
.
o
.
.
cgo
.
il`avi phˆan cu
’
abiˆe
´
nd
ˆo
.
clˆa
.
p,
t´u

.
c l`a theo d
i
.
nh ngh˜ıa: dx =∆x.
D
-
i
.
nh l´y 8.2.1. H`am y = f(x) c´o vi phˆan cˆa
´
p1ta
.
id
iˆe
’
m x
0
khi v`a
chı
’
khi h`am d
´oc´oda
.
oh`amh˜u
.
uha
.
nta
.

id
´ov`aD(x
0
)=f

(x
0
).
Vi phˆan df (x
0
)cu
’
a h`am f ta
.
idiˆe
’
m x
0
biˆe
’
udiˆe
˜
nquada
.
o h`am f

(x
0
)
bo

.
’
i cˆong th´u
.
c
df (x
0
)=f

(x
0
)dx (8.4)
Cˆong th´u
.
c (8.4) cho ph´ep t´ınh vi phˆan cu
’
a c´ac h`am, nˆe
´
ubiˆe
´
td
a
.
o h`am
cu
’
ach´ung.
T`u
.
(8.3) suy ra

y(x
0
+∆x)=y(x
0
)+df (x
0
)+o(dx),dx→ 0.
Nˆe
´
u df (x
0
) =0th`ıdˆe
’
t´ınh gi´a tri
.
gˆa
`
nd´ung cu
’
a h`am f(x)ta
.
idiˆe
’
m
x
0
+∆x ta c´o thˆe
’
´ap du
.

ng cˆong th´u
.
c
y(x
0
+∆x) ≈ y(x
0
)+df (x
0
) (8.5)
Vi phˆan cˆa
´
p 1 c´o c´ac t´ınh chˆa
´
t sau.
1
+
d(αu + βv)=αdu + βdv,
d(uv)=udv + vdu,
d

u
v

=
vdu− udv
v
2
,v=0.
8.2. Vi phˆan 77

2
+
Cˆong th´u
.
c vi phˆan dy = f

(x)dx luˆon luˆon tho
’
a m˜an bˆa
´
t luˆa
.
n
x l`a biˆe
´
nd
ˆo
.
clˆa
.
p hay l`a h`am cu
’
abiˆe
´
ndˆo
.
clˆa
.
p kh´ac. T´ınh chˆa
´

t n`ay
d
u
.
o
.
.
cgo
.
il`at´ınh bˆa
´
tbiˆe
´
nvˆe
`
da
.
ng cu
’
a vi phˆan cˆa
´
p1.
8.2.2 Vi phˆan cˆa
´
p cao
Gia
’
su
.
’

x l`a biˆe
´
nd
ˆo
.
clˆa
.
p v`a h`am y = f(x) kha
’
vi trong lˆan cˆa
.
n n`ao
d
´o c u
’
adiˆe
’
m x
0
. Vi phˆan th´u
.
nhˆa
´
t df = f

(x)dx l`a h`am cu
’
a hai biˆe
´
n

x v`a dx, trong d
´o dx l`a sˆo
´
t`uy ´y khˆong phu
.
thuˆo
.
c v`ao x v`a do d´o
(dx)

=0.
Vi phˆan cˆa
´
p hai (hay vi phˆan th´u
.
hai) d
2
f cu
’
a h`am f(x)ta
.
idiˆe
’
m
x
0
du
.
o
.

.
cd
i
.
nh ngh˜ıa nhu
.
l`a vi phˆan cu
’
a h`am df = f

(x)dx ta
.
idiˆe
’
m x
0
v´o
.
i c´ac d
iˆe
`
ukiˆe
.
n sau dˆay:
1) df pha
’
id
u
.
o

.
.
c xem l`a h`am cu
’
achı
’
mˆo
.
tbiˆe
´
nd
ˆo
.
clˆa
.
p x (n´oi c´ach
kh´ac: khi t´ınh vi phˆan cu
’
a f

(x)dx ta cˆa
`
n t´ınh vi phˆan cu
’
a f

(x), c`on
dx d
u
.

o
.
.
c xem l`a h˘a
`
ng sˆo
´
);
2) Sˆo
´
gia cu
’
abiˆe
´
nd
ˆo
.
clˆa
.
p x xuˆa
´
thiˆe
.
n khi t´ınh vi phˆan cu
’
a f

(x)
d
u

.
o
.
.
c xem l`a b˘a
`
ng sˆo
´
gia d
ˆa
`
u tiˆen, t´u
.
c l`a b˘a
`
ng dx.
Nhu
.
vˆa
.
y theo d
i
.
nh ngh˜ıa ta c´o
d
2
f = d(df )=d(f

(x)dx)=(df


(x))dx = f

(x)dxdx
= f

(x)(dx)
2
hay l`a
d
2
f = f

(x)dx
2
,dx
2
=(dx)
2
. (8.6)
B˘a
`
ng phu
.
o
.
ng ph´ap quy na
.
p, d
ˆo
´

iv´o
.
i vi phˆan cˆa
´
p n ta thu d
u
.
o
.
.
c
cˆong th´u
.
c
d
n
f = f
(n)
(x)dx
n
(8.7)
78 Chu
.
o
.
ng 8. Ph´ep t´ınh vi phˆan h`am mˆo
.
tbiˆe
´
n

Vi phˆan cˆa
´
p n (n>1) cu
’
abiˆe
´
ndˆo
.
clˆa
.
p x du
.
o
.
.
c xem l`a b˘a
`
ng 0, t´u
.
c
l`a
d
n
x =0 v´o
.
i n>1. (8.8)
Nˆe
´
u ∃ d
n

f v`a ∃ d
n
g v`a α, β ∈ R th`ı
d
n
(αf + βg)=αd
n
f + βd
n
g (8.9)
d
n
fg =
n

k=0
C
k
n
d
n−k
f · d
k
g. (8.10)
Ch´u´y. 1) Khi n>1, c´ac cˆong th´u
.
c (8.6) v`a (8.7) chı
’
d
´ung khi x

l`a biˆe
´
nd
ˆo
.
clˆa
.
p. Dˆo
´
iv´o
.
i h`am ho
.
.
p y = y(x(t)) cˆong th´u
.
c (8.6) d
u
.
o
.
.
c
kh´ai qu´at nhu
.
sau:
d
2
y = d(dy)=d(y


x
dx)=d(y

x
)dx + y

x
d(dx)
v`a do d
´o
d
2
y = y

xx
dx
2
+ y

x
d
2
x. (8.11)
Trong tru
.
`o
.
ng ho
.
.

p khi x l`a biˆe
´
nd
ˆo
.
clˆa
.
pth`ıd
2
x = 0 (xem (8.8)) v`a
cˆong th´u
.
c (8.11) tr`ung v´o
.
i (8.6).
2) Khi t´ınh vi phˆan cˆa
´
p n ta c´o thˆe
’
biˆe
´
nd
ˆo
’
iso
.
bˆo
.
h`am d
˜a cho.

Ch˘a
’
ng ha
.
nnˆe
´
u f(x) l`a h`am h˜u
.
uty
’
th`ı cˆa
`
n khai triˆe
’
n n´o th`anh tˆo
’
ng
h˜u
.
uha
.
n c´ac phˆan th´u
.
ch˜u
.
uty
’
co
.
ba

’
n; nˆe
´
u f(x) l`a h`am lu
.
o
.
.
ng gi´ac
th`ı cˆa
`
nha
.
bˆa
.
c nh`o
.
c´ac cˆong th´u
.
cha
.
bˆa
.
c,...
3) T`u
.
cˆong th´u
.
c (8.7) suy ra r˘a
`

ng
f
(n)
(x)=
d
n
f
dx
n
t´u
.
cl`ad
a
.
o h`am cˆa
´
p n cu
’
a h`am y = f(x)ta
.
imˆo
.
tdiˆe
’
m n`ao d´ob˘a
`
ng ty
’
sˆo
´

gi˜u
.
a vi phˆan cˆa
´
p n cu
’
a h`am f(x) chia cho l˜uy th`u
.
abˆa
.
c n cu
’
avi
phˆan cu
’
ad
ˆo
´
isˆo
´
.