DERIVATIVE USING POWER RULE

When we find the derivative of xn, we use the formula

d(xn)/dx = nxn-1

Find dy/dx if :

Problem 1 :

y = x + √x

Solution :

y = x + √x

y = x1 + x1/2

Using the Power rule of derivative,

dy/dx = 1x0 + (1/2) x(1/2 - 1)

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

= 1 + 1/2x1/2

dy/dx = 1 + 1/2√x

Problem 2 :

y = x5 - 3√x

Solution :

y = x5 - 3√x

y = x5 - 3x1/2

Using the Power rule of derivative,

dy/dx = 5x4 - (1/2) 3x(1/2 -1 )

= 5x4 - (1/2) 3x-1/2

= 5x4 - 3/2x1/2

dy/dx = 5x4 - 3/2√x

Problem 3 :

y = x5/2 - 2/x

Solution :

y = x5/2 - 2/x

y = x5/2 - 2x-1

Using the Power rule of derivative,

dy/dx = 5/2 x(5/2 - 1) - 2(-x-1 - 1)

= 5/2 x3/2 + 2x-2

dy/dx = 5/2 x3/2 + 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

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