Sifat-sifat Turunan Fungsi dan Contoh Soal
k = konstanta
n = bilangan
rasional
f’(x) turunan
dari f(x)
u’(x) turunan
dari u(x)
v’(x) turunan dari v(x)
1. Jika f(x) = k maka f’(x) = 0
Contoh soal:
o
f(x) = 9 maka f’(x) = 0
o
f(x) = -20 maka f’(x) = 0
o
f(x) = 1990 maka f’(x) = 0
2. Jika f(x) = x maka f’(x) = 1
Contoh soal:
o
f(x) = 5x maka f’(x)
= 5
o
f(x) = -15x maka f’(x) = -15
o
f(x) = 99x maka f’(x) = 99
3. Jika f(x) = xn maka f’(x) = n xn-1
Contoh soal:
o f(x) = x4
f’(x) = 4x4-1 = 4x3
o f(x) = x99
f’(x) = 99x99-1=
99x98
o f(x) = x5+x3
f’(x) = 5x5-1 +
3x3-1 = 5x4 + 3x2
o f(x) =x6 – (1/x5)
f’(x) = x6 – x-5 = 6x6-1 – (-5)x(-5)-1 = 6x5 + 5x-6 = 6x5 + 5/x6
4. Jika f(x) = axn maka f’(x) = an xn-1
Contoh soal:
o f(x) = 6x2
f’(x) = 6 . 2 . x2-1
= 12x
o f(x) = 15x5
f’(x) = 15 . 5 . x5-1
= 75x4
o f(x) = 20x4 – 3x2 + x
f’(x) = 20 . 4 . x4-1 – 3 . 2 . x2-1 + 1= 80x3 – 6x + 1
5. Jika f(x) = k u(x) maka f’(x) = k u’(x)
Contoh soal:
Tentukan turunan dari f(x) = 3 6x
Jawab:
k = 3
u(x) = 6x maka u’(x) = 6
f’(x) = k u’(x)
f’(x) = 3 . 6 = 18
1 6. Jika f(x) = u(x) ± v(x) maka f’(x) = u’(x) ± v’(x)
Contoh soal:
Tentukan turunan dari f(x) = 3x + 6x2
Jawab:
u(x) = 3x maka u’(x) = 3
v(x) = 6x2 maka v’(x) = 18x
f’(x) = u’(x) ± v’(x)
f’(x) = 3 + 18x
7. Jika f(x) = u(x) . v(x) maka f’(x) = u’(x) . v(x) + u(x) . v’(x)
Contoh soal:
Turunan dari f(x) = (2x+5)(x2+3) yaitu:
Jawab:
u = 2x+5 maka u’(x) = 2
v = x2+3 maka v’ (x) = 2x
f’(x) = u’(x) . v(x) + u(x) . v’(x)
f’(x) = 2(x2 + 3) + (2x + 5) 2x
f’(x) = 2x2 + 6 + 4x2 + 10x
f’(x) =6x2 + 10x + 6
8. Jika f(x) = {u(x)}n maka f’(x) = n {u(x)}n-1 . u’(x)
Contoh soal:
Tentukan turunan dari f(x) = (2x + 1)5
Jawab:
n = 5
u(x) = 2x+1 maka u’(x) = 2
f’(x) = n {u(x)}n-1. u’(x)
f’(x) = 5(2x + 1)5-1 . 2
f’(x) = 10(2x + 1)4
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