The Product and Quotient Rules

The Product Rule

Theorem:

Let f and g be differentiable functions.  Then

          (f(x)g(x))' = f(x)g'(x) + f '(x)g(x)


Proof:

We have 

        d/dx (fg) 

                    f(x+h) g(x+h)  -  f(x) g(x)
        =  lim
                                                    Add and subtract f(x + h)g(x)
                               
        h

                    f(x+h) g(x+h)  -  f(x+h) g(x)  +  f(x+h) g(x)  -  f(x) g(x)
        =  lim
                                                                                               
                               
                            h

                    f(x+h) g(x+h)   -  f(x+h) g(x)               f(x+h) g(x)  -  f(x) g(x)
        =  lim                                                      +                                            
      
                               
      h                                                          h

                               g(x+h) - g(x)                  f(x+h) - f(x)
        =  lim 
f(x+h)                             +                              g(x)   
                               
          h                                 h

        =  [lim f(x+h)] g'(x) + g(x) f '(x)

        =  f(x)g'(x) + g(x)f'(x)

 

Example

Find

            d       
                   (2 - x2) (x4 - 5)
            dx     

Solution:    

Here

        f(x) = 2 - x2 

and

        g(x) = x4 - 5

The product rule gives

       d/dx [f(x)g(x)] = (2 - x2)(4x3) + (-2x)(x4 - 5)



 

The Quotient Rule

Remember the poem

        "lo d hi minus hi d lo square the bottom and away you go"

This poem is the mnemonic for the taking the derivative of a quotient.

Theorem:

          d     f             g f '  - f g'
                                                 
          dx    g                  g2

      


Example:

Find y' if

                     2x - 1
            y =                       
                      x + 1

Solution:

Here

        f(x) = 2x - 1

and

        g(x) = x + 1

The quotient rule gives

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

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

                         3 
           =                             
                     (x + 1)2


Exercise  

Suppose that the cost of producing x snowboards per hour is given by

                        50x + 1000       
C = 100x  +                                 
                            x + 2

find the marginal cost when x  =  10

 

Answer  (hold mouse over yellow rectangle for the answer)

106.25

 


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