#### Transcript Calculus – Session 05 – turunan 02

```Calculus – Session 05
TURUNAN
1
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Turunan / diferensial
 Turunan fungsi f adalah fungsi lain f’ ( dibaca “ f aksen “ ) nilainya
 Contoh : buktikan
limx
2
-1 =2
x1
x–1
= lim (x – 1) ( x + 1)
x1
x–1
= lim
( x + 1)
x1
=1+1
2
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=2
Fungsi turunan disebut derivatif.
f’ (x) =
lim
h0
f (x + h) – f (x)
h
Notasi : f’ (x) = dy/ dx = ∆y/ ∆x = y’
3
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Ex.
Turunan dari f (x) 3x2 + 1 adalah
f ‘ (x) = limf (x + h) – f (x)
h0
h
f (x + h)
= 3(x + h)2 + 1
= 3x2 + 6xh + 3h2 + 1
lim3x2 + 6xh + 3h2 + 1 - 3x2 – 1
Maka, f ’ (x)
=
h0
h
=
6xh +lim
3h2 = 6x + 3h lim
= 6x
h0
h0
h
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Aturan Pencarian Turunan
1.
5
Fungsi Konstan
f (x) = c
turunannya
f’ (x) = 0
2.
Pangkat positif dari x
f (x) = xn
f ‘ (x) = n xn-1
f (x) = axn
f ‘ (x) = a . n xn-1
3.
Pangkat negatif dan pangkat rasional dari x :
f (x) = x-n
f ‘ (x) = -n x-n-1 atau
f ‘ (x) = _-n__
xn+1
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4.
5.
6.
6
Fungsi Identitas
f(x) = x
maka, f’ (x) = 1 yakni
Aturan Jumlah
(f + g )’(x)
= f ’(x) + g ’(x)
D[f (x) + g (x) = Df (x) + Dg (x)
Aturan Selisih
(f - g )’(x)
D[f (x) - g (x)
prepared by eva safaah
= f ’(x) - g ’(x) yakni
= Df (x) - Dg (x)
D(x) = 1
yakni
7.
8.
7
Aturan Hasil Kali
(f . g )’(x)
D[f (x) . g (x)]
= f (x).g’ (x) + g (x). f’ (x)
= f (x). Dg (x) + g (x). Df (x)
Aturan Kelipatan Konstanta
(k f )’(x)
= k . f’ (x)
D [k .f (x)]
= k . Df (x)
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yakni
yakni
9.
8
Aturan Hasil Bagi
‘
f
(x) = g (x) .f ‘(x) - f (x) .g’ (x)
, g (x) ≠ 0
g
g 2(x)
yaitu :
D f (x)
= g (x) .Df (x) - f (x) .Dg (x) , g (x) ≠ 0
g (x)
g 2(x)
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Contoh :
Misalnya 5x2 + 7x – 6
Penyelesainnya :
5d(x2) + 7 d(x) – d(6)
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= 5. 2 x 2-1 + 7 . 1 – 0
= 10x + 7
Contoh :
2x2 + 8x – 3
Penyelesainnya :
2d(x2) + 8 d(x) – d(3) =
=
4x + 8
10
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2. 2 x 2-1 + 8 . 1 – 0
Contoh :
3x2 - 10x
Penyelesainnya :
3d(x2) - 10 d(x) =
3. 2 x 2-1 - 10 . 1
=
6x - 10
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Contoh :
(3x2 – 5)(2x4 – x)
Penyelesaiannya :
D[f (x) . g (x)] = f (x). Dg (x) + g (x). Df (x)
= (3x2 – 5)( 2. 4 x 4-1 – 1 ) + (2x4 – x) ( 3.2x2-1 – 0)
= (3x2 – 5)(8x3 – 1) + (2x4 – x ) ( 6x – 0 )
= (3x2 – 5)(8x3 – 1) + 6x (2x4 – x )
= 24x5 - 3x2 - 40x3 + 5 + 12x5 - 6x2
= 36x5 - 40x3 - 9x2 + 5
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Contoh :
(3x – 5)
(2x2 – 7)
Penyelesaiannya :
D f (x)
= g (x) .Df (x) - f (x) .Dg (x)
g (x)
g 2(x)
= (2x2 – 7)( 3 – 0 ) – (3x – 5)( 4x – 0 )
(2x2 – 7)2
= (2x2 – 7)( 3 ) – (3x – 5)( 4x )
(2x2 – 7)2
Hasil akhir lanjutkan.......
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To be continue ....
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To be continue....
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Latihan
18
1.
Y = (2 – 9x) 15
2.
Y = (5x 2 + 2x -8)5
3.
Y = (4x 3 - 3x 2 + 11x - 1)-5
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Turunan Tingkat Tinggi
Fungsi
f ----> f’
----> f’’ (f dua aksen) -> f’’’ dst..
(turunan ke-2 dari f )
Contoh :
f (x)
=2x3 – 4x2 + 7x – 8
f ‘ (x)
=6x2 – 8x + 7
f ” (x)
=12x – 8
f ‘” (x)
=12
f ” ” (x) = 0
Cause turunan fungsi = 0, maka secara turunan tingkat lebih tinggi
akan “nol”
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Penulisan turunan pertama dari y = f (x) :
f ‘ (x)
DxY
y
Penulisan aksen
penulisan d
Leibniz
Penulisan tingkat tinggi d ( dy )
sebagai
dx
dx
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d2y
dx2
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Latihan
1.
Cari turunan dari f(x) = 2x – 1 dengan definisi turunan
f’ (x) = lim
f (x + h) – f (x)
h0
h
1.
Y = -3x-3
2.
Y = ( 2x – 1 )2
3.
Y =( x2 + 17 )( x4 – 3x + 1 )
4.
Y = x2 – x + 1
3x - 1
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Latihan turunan tingkat tinggi
Cari turunan ke- 2 atau d2y/ dx2 dari fungsi berikut :
1.
F(t) = 1/t
2.
Y = -3x-3
3.
Y = ( 2x – 1 )2
4.
Y =( x2 + 17 )( x4 – 3x + 1 )
5.
Y = x2 – x + 1
3x - 1
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Tugas turunan tingkat tinggi
Cari turunan ke- 3 atau d3y/ dx3 dari fungsi berikut :
1.
Y = ( 2x – 1 )2
2.
Y =( x2 + 17 )( x4 – 3x + 1 )
3.
Y = x2 – x + 1
3x – 1
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To be continue...
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