Carilah turunan pertama dari

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18 November 202118 November 2021

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 . 3x^{5 – 1} – 3 . 12x^{3 – 1} + 1 . 5x^{1 – 1}
y’ = 15x⁴ – 36x² + 5

Jawaban (b) :

y = 2x – 5x² + 7x⁵
y’ = 1 . 2x^{1 – 1} – 2 . 5x^{2 – 1} + 5 . 7 x^{5 – 1}
y’ = 2 – 10x + 35x⁴

Jawaban (c) :

y = $\frac {1} {3}$ x² – $\frac {2} {3}$ x² + 3x
y’ = 2 . $\frac {1} {3}$ x^{2 – 1} – 2 . $\frac {2} {3}$ x^{2 – 1} + 1 . 3x^{1 – 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) = U^1/2 maka y'(U) = 1/2U^-1\2
y’ = U’ . y'(U)
y’ = 2x . 1/2U^-1/2
y’ = x (x² + 5)^-1/2
y’ = $\frac {x} {\sqrt {x^{2} + 5}}$

Tag:Turunan pertama