QUOTIENT RULE FOR DERIVATIVES

Differentiate mentally without simplification:

Problem 1 :

x / (2x - 1)

Solution :

f(x) = x, f’(x) = 0

g(x) = 2x - 1, g’(x) = 2

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= 0(2x - 1) - 2(1) / (2x - 1)²

= -2 / (2x - 1)²

Problem 2 :

x³ / (x² - 4)

Solution :

f(x) = x³, f’(x) = 3x²

g(x) = x² - 4, g’(x) = 2x

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= [3x²(x² - 4) - 2x(x³)] / (x² - 4)²

= [3x⁴ - 12x² - 2x⁴] / (x² - 4)²

= (x⁴ - 12x²) / (x² - 4)²

Problem 3 :

(x + 4) / (x - 6)

Solution :

f(x) = x + 4

f’(x) = 1

g(x) = x - 6

g’(x) = 1

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= 1(x - 6) - 1(x + 4) / (x - 6)²

= (x - 6 - x - 4) / (x - 6)²

= - 10 / (x - 6)²

Problem 4 :

(2x + 5) / (4x - 3)

Solution :

f(x) = 2x + 5, f’(x) = 2

g(x) = 4x - 3, g’(x) = 4

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= [2(4x - 3) - 4(2x + 5)] / (4x - 3)²

= [8x - 6 - 8x - 20] / (4x - 3)²

= -26 / (4x - 3)²

Problem 5 :

x / (2x² - 8)

Solution :

f(x) = x, f’(x) = 1

g(x) = 2x² - 8, g’(x) = 4x

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= 1(2x² - 8) - 4x(x) / (2x² - 8)²

= (2x² - 8 - 4x²) / (2x² - 8)² 

= (- 8 - 2x²) / (2x² - 8)² 

= -(8 + 2x²) / (2x² - 8)² 

Problem 6 :

(x - 7) / x²

Solution :

f(x) = x - 7, f’(x) = 1

g(x) = x², g’(x) = 2x

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= 1(x²) - 2x (x - 7) / (x²)²

= [1(x²) - 2x(x - 7)] / x⁴

= (x² - 2x² + 14x) / x⁴

= (-x² + 14x) / x⁴

= x(-x + 14) / x⁴

= (14 - x) / x³

Problem 7 :

(x² + 4x - 1) / (x + 3)

Solution :

f(x) = x² + 4x - 1

f’(x) = 2x + 4

g(x) = x + 3

g’(x) = 1

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= [(2x + 4) (x + 3) - 1(x² + 4x - 1)] / (x + 3)²

= [(2x² + 6x + 4x + 12) - x² - 4x + 1] / (x + 3)²

= (2x² - x² + 10x - 4x + 12 + 1) / (x + 3)²

= (x² + 6x + 13) / (x + 3)²

Problem 8 :

(x² - 9x + 11) / (2x + 5)

Solution :

f(x) = x² - 9x + 11

f’(x) = 2x - 9

g(x) = 2x + 5

g’(x) = 2

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= (2x - 9) (2x + 5) - 2(x² - 9x + 11) / (2x + 5)²

= (4x² + 10x - 18x - 45 - 2x² + 18x - 22) / (2x + 5)²

= (4x² - 2x² + 10x - 18x + 18x - 45 - 22) / (2x + 5)²

= (2x² + 10x - 67) / (2x + 5)²

Problem 9 :

(3x - 1) / (x² + 12)

Solution :

f(x) = 3x - 1, f’(x) = 3

g(x) = x² + 12, g’(x) = 2x

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= 3(x² + 12) - 2x(3x - 1) / (x² + 12)²

= (3x² + 36 - 6x² + 2) / (x² + 12)²

= (-3x² + 38) / (x² + 12)²

Problem 10 :

(6x + 7) / (x² - x + 3)

Solution :

f(x) = 6x + 7

f’(x) = 6

g(x) = x² - x + 3

g’(x) = 2x - 1

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= [6(x² - x + 3) - (2x - 1) (6x + 7)] / (x² - x + 3)²

= [6x² - 6x + 18 - (12x² + 14x - 6x - 7] / (x² - x + 3)²

= [6x² - 6x + 18 - 12x² + 8x - 7] / (x² - x + 3)²

= (-6x² + 2x + 11) / (x² - x + 3)²

Problem 11 :

(x³ + x) / (x² - x - 1)

Solution :

f(x) = x³ + x

f’(x) = 3x² + 1

g(x) = x² - x - 1

g’(x) = 2x - 1

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= (3x² + 1) (x² - x - 1) - (2x - 1) (x³ + x) / (x² - x - 1)²

= (3x⁴-3x³-3x²+x²-x-1-(2x⁴+2x²-x³-x) / (x² - x - 1)²

= (3x⁴ - 2x⁴ - 3x³ + x³ - 2x² - 2x² - x + x - 1) / (x² - x - 1)²

= (x⁴ - 2x³ - 4x² - 1) / (x² - x - 1)²

Problem 12 :

(5x² - 2x) / (3x + 1)

Solution :

f(x) = 5x² - 2x

f’(x) = 10x - 2

g(x) = 3x + 1

g’(x) = 3

d/dx (f(x) / g(x)) = f’(x) g(x) - g’(x) f(x) / [g(x)]²

= [(10x - 2) (3x + 1) - 3(5x² - 2x)] / (3x + 1)²

= [(30x² + 10x - 6x - 2) - 15x² + 6x] / (3x + 1)²

= [(30x² - 15x² + 10x - 6x + 6x - 2)] / (3x + 1)²

= (15x² + 10x- 2) / (3x + 1)²