1
theresiaveni.wordpress.com
NAMA :
KELAS :
2
theresiaveni.wordpress.com
TURUNAN/DIFERENSIAL
Definisi :
Laju perubahan nilai f terhadap variabelnya adalah :
0
lim)('
h
xfh
xfhxf )()( =
dx
dy
x
y
x
0lim =
dx
df
f (x) merupakan fungsi baru disebut turunan fungsi f atau perbandingan diferensial, proses
mencarinya disebut menurunkan / mendifferensialkan, bagian kalkulus yang berhubungan
dengan itu disebut kalkulus differensial.
f (x) dapat ditulis dengan notasi lain : y atau dx
df atau
dx
dy ( 2 notasi yang terakhir disebut notasi
Leibnitz).
RUMUS-RUMUS TURUNAN
RUMUS-RUMUS TURUNAN
1. Turunan fungsi konstan:
f (x) = k , maka f (x)= 0lim)()(
lim00
h
kk
h
xfhxf
hh
Jadi f (x) = k maka f (x)= 0
2. Turunan fungsi Pangkat
f (x)= xn
maka h
xfhxfxf
h
)()(lim)('
0
= h
xhx nn
h
)(lim
0
= 0
limh h
hxnn
h n .............1.2
)1(nx 221-n
= ....................1.2
)1(lim 21
0
hx
nnnx nn
h
f (x ) = 1nnx
f(x+h)=(x+h) nxxnn ............1.2
)1( 221
hxnn
h nn
f(x) = xn
f(x+h)-f(x) = nx ............1.2
)1( 221
hxnn
h nn
Rumus tsb dibuktikan berlaku untuk n bulat positif ,tetapi ternyata berlaku juga untuk n bulat negatif dan n
pecah. Jadi rumus berlaku untuk n rasional.
3. Turunan Hasil Kali Konstanta Dengan Fungsi
)(.)( xukxf maka h
xfhxfxf
h
)()(lim)('
0
= h
xukhxuk
h
)(.)(.lim
0
=
h
xuhxuk
h
)()(.lim
0
= k.
h
xuhxu
h
)()(.lim
0
)(' xf = )('. xuk
3
theresiaveni.wordpress.com
4. Turunan Jumlah /Selisih Fungsi Fungsi
)()()( xvxuxf maka h
xfhxfxf
h
)()(lim)('
0
= h
xvxuhxvhxu
h
)()()()(lim
0
=
h
xvhxv
h
xuhxu
hh
)()(lim
)()(lim
00
f (x) = u (x) + v (x)
5. Turunan Hasil Kali Fungsi Fungsi
)().()( xvxuxf maka h
xfhxfxf
h
)()(lim)('
0
= h
xvxuhxvhxu
h
)().()()(lim
0
=h
xvxuhxvxuhxvxuhxvhxu
h
)().()()()().()()(lim
0
=
h
xvhxvxu
h
xuhxuhxv
hh
)()()(lim
)()(lim
00
= )(')()(').( xvxuxuxv
= )(')()().(' xvxuxvxu
6. Turunan Hasil Bagi Fungsi Fungsi
Jika f (x) = )(
)(
xv
xu maka u(x) = f (x) v(x)
Menurut rumus sebelumnya u (x) = f (x) v(x) + f (x)v (x)
f (x). v (x) = u (x) - f (x) v (x)
f (x) = )(
)(')(
)()('
xv
xvxv
xuxu
2)(
)(')()()(')('
xv
xvxuxvxuxf
7. Turunan Fungsi Komposisi ( Dalil Rantai )
Jika ))(()( xfgxF maka )('))((')(' xfxfgxF
4
theresiaveni.wordpress.com
Tabel Rumus Turunan dan Contoh:
No f(x) f (x) = y =
=
(turunan pertama dari f(x))
Contoh
1 f(x) = a
dengan a
adalah
konstanta
Turunan Fungsi Konstan
f (x) = 0
f(x) = a f (x) = 0
f(x) = 2 f (x) = 0
f(x) = -100 f (x) = 0
f(x) = 1
2 f (x) = 0
2 f(x) = x Turunan Fungsi Identitas
f (x) = 1
f(x) = 2x f (x) = 2 . (1) = 2
f(x) = -35x f (x) = -35. (1) = -35
f(x) = 3
2 f (x) =
3
2 . (1) =
3
2
3. f(x) = axn Turunan Fungsi Pangkat
f (x) = n . a xn-1
f(x) = axn f (x) = n . a xn-1
f(x) = x2berarti a = 1, n = 2
f (x) = 2. (1) x2-1 = 2x1 = 2x
f(x) = -8x3berarti a = -8, n = 3
f (x) = 3. (-8) x3-1 = -24x2
f(x) = 3
4x4berarti a =
3
4, n = 4
f (x) = 4. (3
4) x4-1 = 3x3
f(x) = 2 = 212berarti a = 2, n =
1
2
f (x) = 1
2. 2
12
1 = 1. 1 2=
12=
1
12
= 1
f(x) = 3
23 =
3
23
= 3 . 2 3
berarti a = 3, n = 2 3
f (x) = (- 2
3). 3
2 3
1 = -2. 5 3=
2
5 3
= 2
53
4. f(x) =
u(x) v(x)
Turunan Jumlah dan
Selisih Fungsi
f (x) = u (x) v (x)
f(x) = u(x) v(x) f (x) = u (x) v (x)
f(x) = -3x+ 4 f (x) = -3 (1) + 0 = -3
f(x) = 12x - 2 f (x) = 12 (1) - 0 = 12
f(x) = 7x2 +3x - 6
f (x) = 2 (7)x2-1 + 3(1) 0 = 14x+ 3
f(x) = - 10x4 + x3 - 2000
f (x) = 4 (-10)x4-1 + 3x3-1 0 = - 40x3 + 3x2
f(x) = x5 - 6 x3 + x
f (x) = 5x4 - 18x2 + 1
f(x) = 2x2 - 4
f (x) = 4x
Ingat:
1
=
1
=
=
= .
5
theresiaveni.wordpress.com
No f(x) f (x) = y =
=
(turunan pertama dari f(x))
Contoh
5. f(x) = u(x) .
v(x)
Turunan Hasil Kali Dua
Fungsi
f(x) = u(x). v(x) + u(x).v (x)
f(x) = u(x).v(x) f(x) = u (x).v(x) + u(x).v(x)
f(x) = (x+ 4)(3x-2)
berarti u(x) = x+ 4 dan v( x) = 3x- 2
u (x) = 1 , v (x) = 3
maka f (x) = u (x) . v(x) + u(x) . v (x)
f(x) = 1(3x- 2) + (x+4)3 = 3x 2 + 3x +12
=6x + 10
Cara lain:
f(x) = (x+ 4)(3x-2) = 3x2 + 10x 8 = 6x + 10
f(x) = (x2+ 2x + 1)(x-1)
berarti u(x) = x2+ 2x +1 dan v( x) = x- 1
u (x) = 2x+2 , v (x) = 1
maka f (x) = u (x) . v(x) + u(x) . v (x)
f (x) = (2x+2 )(x-1) + (x2+ 2x + 1) 1
= 2x2 - 2+ x2+ 2x + 1=3x2+ 2x - 1
6. f(x) = ()
() Turunan Hasil Bagi Dua
Fungsi
f (x) = ().() () . ()
2()
f(x) = ()
() f (x) =
().() () . ()
2()
f(x) = 2
+3 berarti u(x) = 2x dan v( x) = x+3
u (x) = 2 , v (x) = 1
maka f (x) = ().() () . ()
2()
f (x) = 2(+3)2(1)
(+3)2 =
2+62
(+3)2 =
6
2+6+9
f(x) = 4
+1 berarti u(x) = x - 4 dan v( x) = x+1
u (x) = 1 , v (x) = 1
maka f (x) = ().() () . ()
2()
f (x) = 1(+1)(4)(1)
(+1)2 =
+1+4
(+1)2 =
5
2+2+1
7. f(x) = a.(u(x))n Turunan
Aturan/Teorema/Dalil
Rantai
f (x) = n. a. (u(x))n-1 . u (x)
f(x) = (4x 3)5
berarti a = 1, n = 5, u(x) = 4x 3
u (x) = 4 maka f (x) = n. a. (u(x))n-1 . u (x)
f (x) = 5 . 1 (4x 3)5-1. 4
= 5 . 1. 4 (4x 3)4 = 20 (4x 3)4
f(x) = 2(3 + 1)12
berarti a = 2, n =1
2, u(x) = 3x+ 1
u (x) = 3
maka f (x) = n. a. (u(x))n-1 . u (x)
f (x) = 1
2 .2(3 + 1)
12
1. 3
6
theresiaveni.wordpress.com
No f(x) f (x) = y =
=
(turunan pertama dari f(x))
Contoh
= 1. 3(3 + 1) 12 =
3
(3+1)12
= 3
3+1
f(x) = (x+1)(4x 3)5
berarti u(x) = x+ 1 dan v( x) = (4x 3)5
u (x) = 1 , v (x) = 20 (4x 3)4
maka f (x) = u (x) . v(x) + u(x) . v (x)
f (x) = 1(4x 3)5 + (x+1) (20 (4x 3)4)
= (4x 3)4 (( 4x 3) + 20(x+1))
= (4x 3)4 ( 4x 3 + 20x+20)
= (4x 3)4( 24x + 17)
f(x) = (4x 3) 5
+1
berarti u (x) = (4x 3)5 dan v(x) = x +1
u (x) = 20 (4x 3)4 , v (x) = 1
maka f (x) = ().() () . ()
2()
f (x) = 20(4x 3)
4.(+1)(4x 3) 5 .1
(+1)2
= (4x 3)
4.(20+20) (4x 3) 5
2+2+1
=(4x 3)
4 ((20+20) (4x 3))
2+2+1
= (4x 3)
4 (20+204x+3)
2+2+1
= (4x 3)
4 (16+23)
2+2+1
Latihan 1 :
Carilah turunan pertama dari fungsi-fungsi berikut:
Untuk soal no 1 3 tentukan juga turunan kedua fungsi f (x) atau f (x)!
f (x) adalah turunan kedua fungsi f(x) , dapat dicari dengan mencari turunan dari f(x)
1. f (x) = 4x5 - 2x4 + 5x2 x
2. () =3
84
2
33 + 3 + 15
3. f(x) = 4x 2
3 + 4x 1000
4. f(x) = (4x 1)(2x + 3)
5. () = (2 + 1)( + 2)
6. () =25
3
7. f (x) = 32
2
x
x
8. () = (3 4)3
9. () = ( 12)7
10. () = (2 3)( 5)3
11. () = + 2
12. () = 2 3
7
theresiaveni.wordpress.com
13. () =1
3
14. f (x) = x2 + 22
1
x
15. f (x) = 3
3 2 7
xx
16. f(x) =(31
2+5)
2
17. f(x) = 4 + 52 73
18. g(x) = 32
23
19. f(x) = (( +
