Diketahui F(x) = 2/3x^3 + 1/2x^2 – x

Materi :

(1) y = a xⁿ

=> y' = an xⁿ ⁻ ¹

(2) y = a uⁿ

=> y' = an uⁿ ⁻ ¹ . u'

(3) y = u . v

=> y' = u' v + v' u

(4) y = u/v

=> y' = (u' v – v' u)/v²

1) Jika f(x) = 2x³ – 5x² + x – 1, maka f'(x) = ….

Jawab :

f'(x) = 6x² – 10x + 1

2) Diketahui f(x) = 5x² + 4x – 3, nilai f'(2) = ….

Jawab :

f'(x) = 10x + 4

f'(2) = 10(2) + 4 = 24

3) Diketahui f(x) = 3x² – 5x + 2 dan g(x) = x² + 3x – 3. Jika h(x) = f(x) – 2g(x) maka h'(x) = ….

Jawab :

h(x) = f(x) – 2g(x)

h(x) = (3x² – 5x + 2) – 2(x² + 3x – 3)

h(x) = 3x² – 5x + 2 – 2x² – 6x + 6

h(x) = x² – 11x + 8

h'(x) = 2x – 11

4) Turunan pertama dari f(x) = 3x² + x – (1/x) + (2/x²) adalah …

Jawab :

f(x) = 3x² + x – x⁻¹ + 2x⁻²

f'(x) = 6x + 1 + x⁻² – 4x⁻³

f'(x) = 6x + 1 + (1/x²) – (4/x³)

5) Jika f(x) = 3x² – 2ax + 7 dan f'(1) = 0 maka f'(2) = ….

Jawab :

f'(x) = 6x – 2a

f'(1) = 6(1) – 2a = 0

=> -2a = -6

=> a = 3

f'(2) = 6(2) – 2(3) = 12 – 6 = 6

6) f(x) = (3x + 2)(x – 7), f'(x) = …

Jawab :

f(x) = 3x² – 21x + 2x – 14

f(x) = 3x² – 19x – 14

f'(x) = 6x – 19

7) Jika f(x) = 2(3x + 1)⁶, maka f'(x) = …..

Jawab :

f'(x) = 12(3x + 1)⁵ . 3

f'(x) = 36(3x + 1)⁵

8) Jika f(x) = 2(x² – 5x + 2)⁵, maka f'(x) = ….

Jawab :

f'(x) = 10(x² – 5x + 2)⁴ . (2x – 5)

f'(x) = 10(2x – 5)(x² – 5x + 2)⁴

9) Jika f(x) = 3x² (x + 1)³, maka f'(1) = …

Jawab :

u = 3x² => u' = 6x

v = (x + 1)³ => v' = 3(x + 1)² . 1

f'(x) = u' v + v' u

f'(x) = 6x (x + 1)³ + 3(x + 1)² . 3x²

f'(1) = 6(1) (1 + 1)³ + 3(1 + 1)² . 3(1)²

f'(1) = 6 (2)³ + 3(2)² . 3

f'(1) = 48 + 36

f'(1) = 84

10) Jika f(x) = (3x – 5)/(2x + 1), maka f'(x) = ….

Jawab :

u = 3x – 5 => u' = 3

v = 2x + 1 => v' = 2

f'(x) = (u' v – v' u)/v²

f'(x) = (3(2x + 1) – 2(3x – 5))/(2x + 1)²

f'(x) = (6x + 3 – 6x + 10)/(2x + 1)²

f'(x) = 13/(2x + 1)²

11) Jika f(x) = 5/(2x – 1), maka f'(x) = …

Jawab :

f(x) = 5(2x – 1)⁻¹

f'(x) = -5(2x – 1)⁻² . 2

f'(x) = -10/(2x – 1)²

12) Jika y = (x² + 1)(x³ – 1) maka y' = …

Jawab :

y = x⁵ + x³ – x² – 1

y' = 5x⁴ + 3x² – 2x

13) Jika f(x) = px² + 5x – 2 dan f'(1) = 3 maka p = …

Jawab :

f'(x) = 2px + 5

f'(1) = 2p(1) + 5 = 3

=> 2p = -2

=> p = -1

14) Jika f(x) = 8/(3x – 1)², maka f'(x) = …

Jawab :

f(x) = 8(3x – 1)⁻²

f'(x) = -16(3x – 1)⁻³ . 3

f'(x) = -48/(3x – 1)³

15) Jika f(x) = (5x³ – 4x² + 6)/(2x), maka f'(x) = ….

Jawab :

f(x) = (5x³)/(2x) – (4x²)/(2x) + 6/(2x)

f(x) = (5/2)x² – 2x + 3x⁻¹

f'(x) = 5x – 2 – 3x⁻²

f'(x) = 5x – 2 – (3/x²)