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Derivatives of Functions of f(x) = u

 

Introduction

Considering f(x) = (x + x2)5(1 + x3)2, find f'(x)

Letís re-write the function as follows:

y = (x2 + x)5(1 + x3)2

Letís consider a function k as follows:

k  =  x2 + x
k' = 2x  + 1

Letís consider a function h as follows:

u  = k5
   = (x2 + x)5

u' = nkn-1k'
   = 5(x2 + x)4(2x + 1)
   = 5(2x + 1)(x2 + x)4

Letís consider a function g as follows:

g  = 1 + x3

gí = 3x2

Letís consider a function m as follows:

v  = g2
   = (1 + x3)2

v' = ngn-1g'
   = 2(1 + x3)(3x2)
   = 6x2(1 + x3)

Letís consider a function f as follows:

f  = uv

To calculate the derivative, we use the formula (uv)' = u'v + uv':

f' = u'v + uv'
   = 5(2x + 1)(x2 + x)4(1 + x3)2 +(x2 + x)56x2(1 + x3)
   = 5(2x + 1)(x2 + x)4(1 + x3)2 +6x2(1 + x3)(x2 + x)5
   = (x2 + x)4(1 + x3)[5(2x + 1)(1 + x3) + 6x2(x2 + x)]

 

Considering f(x) =

Derivative

, find f'(x)

Letís consider a function k as follows:

g  = 3x3 + 5
g' = 9x2 

Letís consider a function h as follows:

u  = g2
   = (3x3 + 5)2

u' = ngn-1g'
   = 3(3x3 + 5)(9x2)
   = 27x2(3x3 + 5)

Letís consider a function k as follows:

k  = 4x2 + 2x + 3

kí = 8x + 2

Letís consider a function h as follows:

v  = k4 
   = (4x2 + 2x + 3)4
   
ví = nkn-1k'
   = 4(4x2 + 2x + 3)3(8x + 2)
   = 4(8x + 2)(4x2 + 2x + 3)3

Letís consider a function f as follows:

f  = u/v
   = (3x3 + 5)2/(4x2 + 2x + 3)4 

f' = (u'v + uv')/v8 
      
 

Considering f(x) = [3x + (2x + x5)-2]-4, find f'(x)

Letís re-write the function as follows:

y = (3x + (x5 + 2x)-2)-4

Letís consider a function k as follows:

k  =  x5 + 2x
k' = 5x4 + 2

Letís consider a function h as follows:

h  = k-2
   = (x5 + 2x)-2

h' = nkn-1k'
   = -2( x5 + 2x)-3(5x4 + 2)
   = -2(5x4 + 2)(x5 + 2x)-3

Letís consider a function g as follows:

g  = 3x + h
   = 3x + (x5 + 2x)-2

gí = (3x)' + h'
   =  3    + (-2(5x4 + 2)(x5 + 2x)-3)
   = -2(5x4 + 2)(x5 + 2x)-3 + 3

Letís consider a function f as follows:

f  = g-4
   = [3x + (x5 + 2x)-2]-4

f' = ngn-1g'
   = -4[3x + (x5 + 2x)-2]-5[-2(5x4 + 2)(x5 + 2x)-3 + 3]
   = -4[-2(5x4 + 2)(x5 + 2x)-3 + 3][3x + (x5 + 2x)-2]-5

 

 

 

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