DERIVATIVE USING POWER RULE

Find dy/dx if :

Problem 1 :

y = x + √x

Solution :

y = x + √x

y = x¹ + x^1/2

Using the Power rule of derivative,

dy/dx = 1x⁰ + (1/2) x^{(1/2 - 1)}

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

= 1 + 1/2x^1/2

dy/dx = 1 + 1/2√x

Problem 2 :

y = x⁵ - 3√x

Solution :

y = x⁵ - 3√x

y = x⁵ - 3x^1/2

Using the Power rule of derivative,

dy/dx = 5x⁴ - (1/2) 3x^{(1/2 -1 )}

= 5x⁴ - (1/2) 3x^-1/2

= 5x⁴ - 3/2x^1/2

dy/dx = 5x⁴ - 3/2√x

Problem 3 :

y = x^5/2 - 2/x

Solution :

y = x^5/2 - 2/x

y = x^5/2 - 2x^-1

Using the Power rule of derivative,

dy/dx = 5/2 x^{(5/2 - 1)} - 2(-x^{-1 - 1})

= 5/2 x^3/2 + 2x^-2

dy/dx = 5/2 x^3/2 + 2/x²

Problem 4 :

y = 3x⁴ - 2/x + 6/x²

Solution :

y = 3x⁴ - 2/x + 6/x²

y = 3x⁴ - 2x^-1 + 6x^-2

Using the Power rule of derivative,

dy/dx = 12x³ - 2(-x^{-1 -1} ) + (-12x^{-2 - 1})

= 12x³ + 2x^-2 - 12x^-3

dy/dx = 12x³ + 2/x² - 12/x³

Problem 5 :

y = (x + 5) (x + 2)

Solution :

y = (x + 5) (x + 2)

y = x² + 2x + 5x + 10

y = x² + 7x + 10

Using the Power rule of derivative,

dy/dx = 2x + 7

Problem 6 :

y = (3x + 1) (5x - 3)

Solution :

y = (3x + 1) (5x - 3)

y = 15x² - 9x + 5x - 3

y = 15x² - 4x - 3

Using the Power rule of derivative,

dy/dx = 30x - 4

Problem 7 :

y = (5x² - 3) (4x³ + x)

Solution :

y = (5x² - 3) (4x³ + x)

y = 20x⁵ + 5x³ - 12x³ - 3x

Using the Power rule of derivative,

dy/dx = 100x⁴ + 15x² - 36x² - 3

dy/dx = 100x⁴ - 21x² - 3

Problem 8 :

y = (x³ + 1) (2x + 3)

Solution :

y = (x³ + 1) (2x + 3)

y = 2x⁴ + 3x³ + 2x + 3

Using the Power rule of derivative,

dy/dx = 8x³ + 9x² + 2

Problem 9 :

y = (x⁵ - 2x)²

Solution :

y = (x⁵ - 2x)²

y = (x⁵ - 2x) (x⁵ - 2x)

y = x¹⁰ - 2x⁶ - 2x⁶ + 4x²

y = x¹⁰ - 4x⁶ + 4x²

Using the Power rule of derivative,

dy/dx = 10x⁹ - 24x⁵ + 8x

Problem 10 :

y = (x - 2) (x + 1) (3x + 1)

Solution :

y = (x - 2) (x + 1) (3x + 1)

y = x(x + 1) (3x + 1) - 2(x + 1) (3x + 1)

y = x(3x² + x + 3x + 1) - 2(3x² + x + 3x + 1)

y = 3x³ + x² + 3x² + x - 6x² - 2x - 6x - 2

y = 3x³ - 2x² - 7x - 2

Using the Power rule of derivative,

dy/dx = 9x² - 4x - 7

Problem 11 :

y = (x - a)³

Solution :

y = (x - a)³

By using binomial formula,

(a - b)³ = a³ - 3a²b + 3ab² - b³

y = x³ - 3x²a + 3xa² - a³

Using the Power rule of derivative,

dy/dx = 3x² - 6xa + 3a²

Problem 12 :     

y = (2x + 3)³

Solution :

y = (2x + 3)³

By using binomial formula,

(a + b)³ = a³ + 3a²b + 3ab² + b³

y = (2x)³ + 3(2x)²(3) + 3(2x)(3)² + 3³

y = 8x³ + 36x² + 54x + 27

Using the Power rule of derivative,

dy/dx = 24x² + 72x + 54

Problem 13 :

y = 2x (3x² - 7x + 8)

Solution :

y = 2x (3x² - 7x + 8)

y = 6x³ - 14x² + 16x

Using the Power rule of derivative,

dy/dx = 18x² - 28x + 16

Problem 14 :

y = 3x² (x + 1) (x - 2)

Solution :

y = 3x² (x + 1) (x - 2)

y = 3x² (x² - 2x + x - 2)

y = 3x² (x² - x - 2)

y = 3x⁴ - 3x³ - 6x²

Using the Power rule of derivative,

dy/dx = 12x³ - 9x² - 12x

Problem 15 :

y = (x + 1/x)²

Solution : 

y = (x + 1/x)²

By using binomial formula,

(a + b)² = a² + b² + 2ab

y = x² + (1/x)² + 2x(1/x)

y = x² + 1/x² + 2

y = x² + x^-2 + 2

Using the Power rule of derivative,

dy/dx = 2x + (-2)x^{(-2 - 1)}

= 2x - 2x^-3

dy/dx = 2x - 2/x³