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Derivatives of f(x) = u

 

Introduction

Considering f(x) = (2 - 5x)3, find f'(x)

Letís re-write the function as follows:

y = (-5x + 2)3

Let's consider the function

u  = -5x + 2
u' = -5

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

un    = (-5x + 2)3
(un)' = nun-1u'
      = 3(-5x + 2)2(-5)
      = -15(-5x + 2)2

Considering f(x) = (7 - 2x3)-4, find f'(x)

Letís re-write the function as follows:

y = (-2x3 + 7)-4

Let's consider the function u as follows

u  = -2x3 + 7
u' = -6x2

Let's consider the function u as follows

f  = u-4

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

un    = (-2x3 + 7)-4
(un)' = nun-1u'
      = -4(-2x3 + 7)-5(-6x2)
      = -24x2(7 - 2x3)-5

Find the derivative of

Let's consider

u  = 4x - 7

u' = 4

Remember the formula to find the derivative of the inverse of a function:

f'(x) = 
 

Find the derivative of

Let's consider

u  = x - 1

u' = 1

And let's consider

v  = x + 1

v' = 1

Remember the formula to find the derivative of the division of two functions:

       
       
    
 

Considering f(x) =

find f'(x)

Letís re-write the function as follows:

y = (2x + 1)-3

Let's consider the function

u  = 2x + 1
u' = 2

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

un    = (2x + 1)-3
(un)' = nun-1u'
      = -3(2x + 1)-4(2)
      = -6(2x + 1)-4
       =
Derivative
 
 

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