USING CHAIN RULE WITH POLYNOMIAL FUNCTIONS

Problem 1 : 

(x - x² - x⁴)⁵ / 5

Solution :

= (x - x² - x⁴)⁵ / 5

= (1/5) 5(x - x² - x⁴)⁴ d/dx (x - x² - x⁴)

= (x - x² - x⁴)⁴ (1 - 2x - 4x³)

Problem 2 :

(5 - x)^-2 / 2

Solution :

= (5 - x)^-2 / 2

= 1/2 (-2) (5 - x)^-3 · d/dx (5 - x)

= - (5 - x)^-3 (0 - 1)

= (5 - x)^-3

Problem 3 :

(2x + 1)^-1

Solution :

= (2x + 1)^-1

= -1(2x + 1)^-2 d/dx (2x + 1)

= - (2x + 1)^-2 (2)

= -2(2x + 1)^-2

Problem 4 :

(9 - 4x)^-3

Solution :

= (9 - 4x)^-3

= -3(9 - 4x)^-4 d/dx (9 - 4x)

= -3(9 - 4x)^-4 (0 - 4)

= 12 (9 - 4x)^-4

Problem 5 :

(4x² - 3x³ + x)^-2

Solution :

= (4x² - 3x³ + x)^-2

= -2(4x² - 3x³ + x)^-3 d/dx (4x² - 3x³ + x)

= -2(4x² - 3x³ + x)^-3 · (8x - 9x² + 1)

= -2(8x - 9x² + 1) / (4x² - 3x³ + x)³

Problem 6 :

5(x² - 9)^-3

Solution :

= 5(x² - 9)^-3

= 5 (-3)(x² - 9)^-4 d/dx (x² - 9)

= -15(x² - 9)^-4 (2x)

= -30x (x² - 9)^-4

Problem 7 :

(x^-1 - 2x^-2)^-3

Solution :

= (x^-1 - 2x^-2)^-3

= -3(x^-1 - 2x^-2)^-4 d/dx (x^-1 - 2x^-2)

= -3(x^-1 - 2x^-2)^-4 · (-x^-2 - 2(-2)x^-3)

= -3(x^-1 - 2x^-2)^-4 · (-x^-2 + 4x^-3)

= -3x⁵ (4 - x) (x - 2)^-4

Problem 8 :

(3x + 1)^1/2

Solution :

= (3x + 1)^1/2

= (1/2) (3x + 1)^-1/2 d/dx (3x + 1)

= (1/2) (3x + 1)^-1/2 (3)

= 3 / 2√(3x + 1)

Problem 9 :

(6x + 1)^1/3

Solution :

= (6x + 1)^1/3

= (1/3)(6x + 1)^-2/3 d/dx (6x + 1)

= (1/3)(6x + 1)^-2/3 (6)

= 2(6x + 1)^-2/3

Problem 10 :

(5x + 7)^3/2

Solution :

= (5x + 7)^3/2

= (3/2) (5x + 7)^1/2 d/dx (5x + 7)

= (3/2) (5x + 7)^1/2 (5)

= 15(5x + 7)^1/2 / 2

Problem 11 :

(x³ - 5x² + x)^3/4

Solution :

= (x³ - 5x² + x)^3/4

= (3/4) (x³ - 5x² + x)^-1/4 d/dx (x³ - 5x² + x)

= (3/4) (x³ - 5x² + x)^-1/4 (3x² - 10x + 1)

= 3(3x² - 10x + 1) / 4∜ (x³ - 5x² + x)

Problem 12 :

(x⁵ - 5x)^1/5

Solution :

= (x⁵ - 5x)^1/5

= (1/5) (x⁵ - 5x)^-4/5 d/dx (x⁵ - 5x)

= (1/5) (x⁵ - 5x)^-4/5 (5x⁴ - 5)

= 5(1/5) (x⁵ - 5x)^-4/5 (x⁴ - 1)

= (x⁴ - 1) (x⁵ - 5x)^-4/5