Question

Tentukan turunan pertama dari fungsi di bawah ini dan nyatakan dalam bentuk perkalian faktor
f(x) = (3x - 2)⁵ (x² - 3x + 2)³

Answer 1

Verified answer

Jawab:
= 3(3x-2)⁴(x-2)²(x-1)²(11x²-28x+16)

Penjelasan:
f(x) = (3x - 2)⁵ (x² - 3x + 2)³
Tentukan turunan pertama

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

Maka jika
f(x) = (3x - 2)⁵ (x² - 3x + 2)³
f'(x) = [(3x - 2)⁵]'(x² - 3x + 2)³ + (3x - 2)⁵[(x² - 3x + 2)³]'
f'(x) = [(3x - 2)⁵]'(x² - 3x + 2)³ + (3x - 2)⁵[(x² - 3x + 2)³]'

Aturan rantai
[(f(x))ⁿ]'  =  n(f(x))ⁿ⁻¹(f(x))'

Untuk [(3x - 2)⁵]'
[(3x - 2)⁵]' = 5(3x-2)⁵⁻¹(3x-2)'
[(3x - 2)⁵]' = 5(3x-2)⁴(3)
[(3x - 2)⁵]' = 15(3x-2)⁴

Untuk [(x² - 3x + 2)³]'
[(x² - 3x + 2)³]' = 3(x²-3x+2)³⁻¹(x²-3x+2)'
[(x² - 3x + 2)³]' = 3(x²-3x+2)²(2x-3)
[(x² - 3x + 2)³]' = (6x-9)(x²-3x+2)²

Maka
f'(x) = 15(3x-2)⁴(x² - 3x + 2)³ + (3x-2)⁵(6x-9)(x²-3x+2)²
f'(x) = (3x-2)⁴(15(x² - 3x + 2)³ + (3x-2)(6x-9)((x-2)(x-1))²)
f'(x) = (3x-2)⁴(15((x-2)(x-1))³ + (3x-2)(6x-9)(x-2)²(x-1)²)
f'(x) = (3x-2)⁴(15(x-2)³(x-1)³ + (3x-2)(6x-9)(x-2)²(x-1)²)
f'(x) = (3x-2)⁴(x-2)²(x-1)²(15(x-2)(x-1) + (3x-2)(6x-9))
f'(x) = (3x-2)⁴(x-2)²(x-1)²(15(x-2)(x-1) + (3x-2)(6x-9))
f'(x) = (3x-2)⁴(x-2)²(x-1)²(3(5(x-2)(x-1)) + 3(3x-2)(2x-3))
f'(x) = (3x-2)⁴(x-2)²(x-1)²(3(5(x-2)(x-1)+(3x-2)(2x-3)))
f'(x) = (3x-2)⁴(x-2)²(x-1)²(3(5(x-2)(x-1)+(3x-2)(2x-3)))
f'(x) = 3(3x-2)⁴(x-2)²(x-1)²(5(x-2)(x-1)+(3x-2)(2x-3))
f'(x) = 3(3x-2)⁴(x-2)²(x-1)²(5(x²-3x+2)+6x²-13x+6)
f'(x) = 3(3x-2)⁴(x-2)²(x-1)²(5x²-15x+10+6x²-13x+6)
f'(x) = 3(3x-2)⁴(x-2)²(x-1)²(11x²-28x+16)

(xcvi)