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Carilah turunan pertama dari

Contoh Soal 1

Carilah turunan pertama dari :
a. y = 3x5 – 12x3 + 5x
b. y = 2x – 5x2 + 7x5
c. y = \frac {1} {3} x2\frac {2} {3} x2 + 3x

Pembahasan

Jawaban (a) :

  • y = 3x5 – 12x3 + 5x
  • y’ = 5 . 3x5 – 1 – 3 . 12x3 – 1 + 1 . 5x1 – 1
  • y’ = 15x4 – 36x2 + 5

Jawaban (b) :

  • y = 2x – 5x2 + 7x5
  • y’ = 1 . 2x1 – 1 – 2 . 5x2 – 1 + 5 . 7 x5 – 1
  • y’ = 2 – 10x + 35x4

Jawaban (c) :

  • y = \frac {1} {3} x2\frac {2} {3} x2 + 3x
  • y’ = 2 . \frac {1} {3} x2 – 1 – 2 . \frac {2} {3} x2 – 1 + 1 . 3x1 – 1
  • y’ = \frac {2} {3}x – \frac {4} {3} x + 3

Contoh Soal 2

Carilah turunan pertama dari:
a. y = (x + 2) (2x – 7)
b. y = (3x + 4) (5x – 2)
c. y = (5x + 2) (x2 – 3)

Pembahasan

Jawaban (a) :

  • y = (x + 2) (2x – 7)
  • U = x + 2 maka U’ = 1
  • V = 2x – 7 maka V’ = 2
  • y = U . V
  • y’ = U’ . V + U . V’
  • y’ = 1 . (2x – 7) + (x + 2) . 2
  • y’ = (2x – 7) + (2x + 4)
  • y’ = 2x + 2x – 7 + 4 = 4x – 3

Jawaban (b) :

  • y = (3x + 4) (5x – 2)
  • U = 3x + 4 maka U’ = 3
  • V = 5x – 2 maka V’ = 5
  • y = U . V
  • y’ = U’ . V + U . V’
  • y’ = 3 (5x – 2) + (3x + 4) . 5
  • y’ = 15x – 6 + 15x + 20
  • y’ = 30x + 24

Jawaban (c):

  • y = (5x + 2) (x2 – 3)
  • U = 5x + 2 maka U’ = 5
  • V = x2 – 3 maka V’ = 2x
  • y = U . V
  • y’ = U’ . V + U . V’
  • y’ = 5 (x2 – 3) + (5x + 2) . 2x
  • y’ = 5x2 – 15 + 10x2 + 4x
  • y’ = 15x2 + 4x – 15

Contoh soal 3

Carilah turunan pertama dari:
a. y = \frac {x - 5} {4x + 2}
b. y = \frac {2 - 5x} {x + 2}
c. y = \frac {x^{2} + 1} {1 - x}

Pembahasan

Jawaban (a) :

  • y = \frac {x - 5} {4x + 2}
  • U = x – 5 maka U’ = 1
  • V = 4x + 2 maka V’ = 4
  • y = \frac {U} {V}
  • y’ = \frac {U' . V - U . V'} {V^2}
  • y’ = \frac {1 (4x + 2) - (x - 5) . 4} {(4x + 2)^2}
  • y’ = \frac {4x + 2 - 4x + 20} {(4x + 2)^2}
  • y’ = \frac {22} {(4x + 2)^2}

Jawaban (b) :

  • y = \frac {2 - 5x} {x + 2}
  • U = 2 – 5x maka U’ = -5
  • V = x + 2 maka V’ = 1
  • y = \frac {U} {V}
  • y’ = \frac {U' . V - U . V'} {V^2}
  • y’ = \frac {-5 (x + 2) - (2 - 5x) . 1} {(x + 2)^2}
  • y’ = \frac {-5x - 10 - 2 + 10x} {(x + 2)^2}
  • y’ = \frac {5x - 12} {(x + 2)^2}

Jawaban (c) :

  • y = \frac {x^{2} + 1} {1 - x}
  • U = x2 + 1 maka U’ = 2x
  • V = 1 – x maka V’ = -1
  • y = \frac {U} {V}
  • y’ = \frac {U' . V - U . V'} {V^2}
  • y’ = \frac {2x (1 - x) - (x^{2} + 1) . -1} {(1 - x)^2}
  • y’ = \frac {2x - 2x^{2} + x^{2} + 1} {(1 - x)^2}
  • y’ = \frac {-x^{2} + 2x + 1} {(1 - x)^2}

Contoh soal 4

Carilah turunan pertama dari :
a. y = (2x + 3)3
b. y = (2 – x)5
c. y = \sqrt {x^{2} + 5}

Pembahasan

Jawaban (a) :

  • y = (2x + 3)3
  • U = 2x + 3 maka U’ = 2
  • y(U) = U3 maka y'(U) = 3U2
  • y’ = U’ . y'(U)
  • y’ = 2 . 3U2
  • y’ = 6 (2x + 3)2

Jawaban (b) :

  • y = (2 – x)5
  • U = 2 – x maka U’ = -1
  • y(U) = U5 maka y'(U) = 5U4
  • y’ = U’ . y'(U)
  • y’ = -1 . 5U4
  • y’ = -5 (2 – x)4

Jawaban (c) :

  • y = \sqrt {x^{2} + 5}
  • y = (x2 + 5)1/2
  • U = x2 + 5 maka U’ = 2x
  • y(U) = U1/2 maka y'(U) = 1/2U-1\2
  • y’ = U’ . y'(U)
  • y’ = 2x . 1/2U-1/2
  • y’ = x (x2 + 5)-1/2
  • y’ = \frac {x} {\sqrt {x^{2} + 5}}

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