PRODUCT RULE FOR DERIVATIVES

Differentiate on paper without simplification:

Problem 1 :

(x + 2) (x² - 2x + 7)

Solution :

f(x) = x + 2, f’(x) = 1

g(x) = x² - 2x + 7, g’(x) = 2x - 2

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= 1(x² - 2x + 7) + (x + 2) (2x - 2)

= x² - 2x + 7 + 2x² - 2x + 4x - 4

= x² - 2x + 7 + 2x² + 2x - 4

= 3x² + 3

Problem 2 :

(1 - x³) (7x + 4)

Solution :

f(x) = 1 - x³, f’(x) = -3x²

g(x) = 7x + 4, g’(x) = 7

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= -3x² (7x + 4) + (1 - x³) (7)

= -21x³ - 12x² + 7 - 7x³

= -28x³ - 12x² + 7

Problem 3 :

(3x - 5) (x³ + 2x² - 8)

Solution :

f(x) = 3x - 5 ,f’(x) = 3

g(x) = x³ + 2x² - 8, g’(x) = 3x² + 4x

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= 3(x³ + 2x² - 8) + (3x - 5) (3x² + 4x)

= 3x³ + 6x² - 24 + 9x³ + 12x² - 15x² - 20x

= 12x³ + 3x² - 20x - 24

Problem 4 :

(x² - 2) (5x - x³)

Solution :

f(x) = x² - 2, f’(x) = 2x

g(x) = 5x - x³, g’(x) = 5 - 3x²

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= 2x (5x - x³) + (x² - 2) (5 - 3x²)

= 10x² - 2x³ + 5x² - 3x⁴ - 10 + 6x²

= - 3x⁴ - 2x³ + 16x² + 5x² - 10

= - 3x⁴ - 2x³ + 21x² - 10

Problem 5 :

(x² + 3x - 1) (x³ - 4x + 7)

Solution :

f(x) = x² + 3x - 1, f’(x) = 2x + 3

g(x) = x³ - 4x + 7, g’(x) = 3x² - 4

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= (2x + 3) (x³ - 4x + 7) + (x² + 3x - 1) (3x² - 4)

= 2x⁴ - 8x + 14x + 3x³ - 12x + 21 + 3x⁴ - 4x² + 9x³ - 12x - 3x² + 4 

= 7x⁴ + 12x³ - x² - 18x + 25

Problem 6 :

(x³ - 2x + 8) (6 - 5x)

Solution :

f(x) = x³ - 2x + 8, f’(x) = 3x² - 2

g(x) = 6 - 5x, g’(x) = -5

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= (3x² - 2) (6 - 5x) + (x³ - 2x + 8) (-5)

= 18x² - 15x³ - 12 + 10x - 5x³ + 10x - 40

= 18x² - 20x³ + 20x - 52

= - 20x³ + 18x² + 20x - 52

Problem 7 :

(8x² - 5x) (13x² - 4)

Solution :

f(x) = 8x² - 5x, f’(x) = 16x - 5

g(x) = 13x² - 4, g’(x) = 26x

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= (16x - 5) (13x² - 4) + (8x² - 5x) (26x)

= 208x² - 64x - 45x² + 20 + 208x³ - 130x

= 208x³ - 163x² - 194x + 20

Problem 8 :

(x⁵ - 2x³) (7x² + x - 8)

Solution :

f(x) = x⁵ - 2x³, f’(x) = 5x⁴ - 6x²

g(x) = 7x² + x - 8, g’(x) = 14x + 1

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= (5x⁴ - 6x²) (7x² + x - 8) + (x⁵ - 2x³) (14x + 1)

= 35x⁶ + 5x⁵ - 40x⁴ - 42x⁴ - 6x³ + 48x² + 14x⁶ + x⁵ - 28x⁴ - 2x³

= 49x⁶ + 6x⁵ - 110x⁴ - 4x³ + 48x²

Problem 9 :

(3 - x³) (8x + 1)

Solution :

f(x) = 3 - x³, f’(x) = -3x²

g(x) = 8x + 1, g’(x) = 8

d/dx (f(x) · g(x)) = f’(x) · g(x) + f(x) · g’(x)

= (-3x²) (8x + 1) + (3 - x³) (8)

= - 24x³ - 3x² + 24 - 8x³

= - 32x³ - 3x² + 24