113
dx du
u csch
u coth
dx du
. du
dy dx
dy -
= =
terbukti
Contoh 4.30
Jika y = -3csch
x 2
1 5
1 +
, tentukan
dt dy
Penyelesaian : Misal : u =
x 2
1 5
1 +
y = -3 csch u
2 1
dx du
= cschu
u coth
3 du
dy =
x 2
1 5
1 sech
x 2
1 5
1 coth
2 3
2 1
cschu u
coth 3
dt du
du dy
dt dy
+ +
= =
=
Soal-soal Tentukan turunan pertama dari fungsi-fungsi berikut
1. y = sinh2-3x 6. y =
x 2
1 coth
c bx
ax
2
+ +
+
2. y = cosha
2
x – b 7. y =
x 2
h sec
e
ax -
3. y = x
2
sinh5x 8. y =
x 5
4 ln
x 3
h sec
-
4. y = e
mx
cosh2x 9. y =
1 -
cschx x
5 1
3
5. y = ln2-x tanh3x 10. y =
bx -
cscha e
x 3
1
4.11 Turunan fungsi hiperbolik invers
Jika y = fx = sinh
-1
x maka
= =
x f
dx dy
1 x
1
2
+
4.56 Bukti : y = fx = sinh
-1
x =
1 x
x ln
2
+ +
1 x
1 1
x x
1 .
1 x
x 1
x 1
x x
1 x
x 1
dx dy
2 2
2 2
2 2
+ =
+ +
+ +
+ =
+ +
+ +
=
terbukti
Jika y = sinh
-1
u dan u = fx maka
= dx
dy dx
du 1
u 1
2
+
4.57 Bukti : y = sinh
-1
u ®
1 u
1 du
dy
2
+ =
dx du
1 u
1 dx
du .
du dy
dx dy
2
+ =
=
terbukti
114
Contoh 4.31
Jika y = -3sinh
-1
x 2
1
, tentukan
dt dy
Penyelesaian : Misal : u =
x 2
1
y = -3 sinh
-1
u
2 1
dx du
=
1 1
3
2
+ -
= u
du dy
1 4
1 2
3 2
1 1
1 3
2 2
+ -
= +
- =
= x
u dt
du du
dy dt
dy
Jika y = fx = cosh
-1
x maka
= =
x f
dx dy
1 x
1
2
-
, x 1 4.58
Bukti : y = fx = cosh
-1
x =
1 x
x ln
2
- +
1 x
1 1
x x
1 .
1 x
x 1
x 1
x x
1 x
x 1
dx dy
2 2
2 2
2 2
- =
- +
- +
- =
- +
- +
=
, x 1 terbukti
Jika y = cosh
-1
u dan u = fx maka
= dx
dy dx
du 1
u 1
2
-
, u 1 4.59
Bukti : y = cosh
-1
u ®
1 u
1 du
dy
2
- =
dx du
1 u
1 dx
du .
du dy
dx dy
2
- =
=
, u 1 terbukti
Contoh 4.32
Jika y = cosh
-1
x 4
3
, tentukan
dx dy
Penyelesaian : Misal : u =
x 4
3
y = cosh
-1
u
4 3
dx du
= 1
u 1
du dy
2
- =
1 x
16 9
4 3
4 3
1 u
1 dt
du du
dy dt
dy
2 2
+ =
- =
=
Jika y = fx = tanh
-1
x maka
= =
x f
dx dy
2
x 1
1 -
, 1
x 4.60
Bukti : y = fx = tanh
-1
x =
1 x
, x
1 x
1 ln
2 1
- +
115 2
2
x 1
1 x
1 x
1 .
x 1
2 2
1 dx
dy -
= +
- -
=
,
1 x
terbukti
Jika y = tanh
-1
u dan u = fx maka
= dx
dy dx
du u
1 1
2
-
,
1 u
4.61 Bukti : y = tanh
-1
u ®
2
u 1
1 du
dy -
= dx
du u
1 1
dx du
. du
dy dx
dy
2
- =
=
, 1
u terbukti
Contoh 4.33
Jika y = tanh
-1
2x-1, tentukan
dx dy
Penyelesaian : Misal : u = 2x - 1
y = tanh
-1
u
2 dx
du =
2
u 1
1 du
dy -
=
2 2
1 x
2 1
2 2
u 1
1 dx
du du
dy dx
dy -
- =
- =
=
Jika y = fx = coth
-1
x maka
= =
x f
dx dy
2
x 1
1 -
,
1 x
4.62 Bukti : y = fx = tanh
-1
x =
1 x
, 1
x 1
x ln
2 1
- +
2 2
2
x 1
1 1
x 1
1 x
1 x
. 1
x 2
2 1
dx dy
- =
- -
= +
- -
- =
,
1 x
terbukti Jika y = coth
-1
u dan u = fx maka
= dx
dy dx
du u
1 1
2
-
,
1 u
4.63 Bukti : y = tanh
-1
u ®
2
u 1
1 du
dy -
= dx
du u
1 1
dx du
. du
dy dx
dy
2
- =
=
,
1 u
terbukti
Contoh 4.34
Jika y = 3 coth
-1
2-3x, tentukan
dx dy
Penyelesaian : Misal : u = 2-3x
y = 3 tanh
-1
u
3 dx
du -
=
2
u 1
3 du
dy -
=
2 2
x 3
2 1
9 3
u 1
3 dx
du du
dy dx
dy -
- -
= -
- =
=
116
Jika y = fx = sech
-1
x maka
= =
x f
dx dy
2
x 1
x 1
- -
,
1 x
4.64 Bukti : y = fx = sech
-1
x =
1 x
, x
x 1
1 ln
2
- +
2
x 1
x 1
dx dy
- -
=
,
1 x
terbukti
Jika y = sech
-1
u dan u = fx maka
= dx
dy dx
du u
1 u
1
2
- -
, 1
u 4.65
Bukti : y = sech
-1
u ®
2
u 1
u 1
du dy
- -
= dx
du u
1 1
dx du
. du
dy dx
dy
2
- -
= =
,
1 u
terbukti
Contoh 4.35
Jika y = -2 sech
-1
1-x, tentukan
dx dy
Penyelesaian : Misal : u = 1-x
y = 2 sech
-1
u
1 dx
du -
=
2
u 1
u 2
du dy
- -
=
2 2
x 1
1 x
1 2
1 u
1 u
2 dx
du du
dy dx
dy -
- -
= -
- -
= =
Jika y = fx = csch
-1
x maka
= =
x f
dx dy
2
x 1
x 1
+ -
4.66 Bukti : y = fx = csch
-1
x =
x x
1 1
ln
2
+ +
2
x 1
x 1
dx dy
+ -
=
terbukti
Jika y = csch
-1
u dan u = fx maka
= dx
dy dx
du u
1 u
1
2
+ -
4.67 Bukti : y = csch
-1
u ®
2
u 1
u 1
du dy
+ -
= dx
du u
1 u
1 dx
du .
du dy
dx dy
2
+ -
= =
terbukti
Contoh 4.36
117
Jika y = csch
-1
sinx, tentukan
dx dy
Penyelesaian : Misal : u = sinx
y = csch
-1
u
x cos
dx du
=
2
u 1
u 1
du dy
+ -
=
x sin
1 x
sin x
cos x
cos u
1 u
1 dx
du du
dy dx
dy
2 2
+ -
= +
- =
=
Soal-soal Tentukan turunan pertama dari fungsi-fungsi :
1. y = sinh
-1
cosx 4. y = x
2
coth
-1
x 2. y = cosh
-1
sin2x 5. y = sech
-1
x sinx 3. y = tanh
-1
3x+p 6. y = e
-2x
csch
-1
1-2x
4.12 Turunan tingkat tinggi
