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Mathematics LibreTexts

The Product and Quotient Rules

The Product Rule

Theorem  (The Product Rule)

Let f and g be differentiable functions.  Then

\[  \left[f(x) \, g(x)\right] ' = f(x)\, g '(x) + f '(x) \,g(x) \]


Proof:

We have 

       

Example \(\PageIndex{1}\):

Find

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

Solution:    

Here

        f(x)  =  2 - x2 

and

       g(x)  =  x4 - 5

The product rule gives

          d
               (2 - x2)(x4 - 5)  =  (2 - x2)(4x3) + (-2x)(x4 - 5)
        dx

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

\[ \dfrac{d}{dx} \dfrac{f(x)}{g(x)} = \dfrac{g(x)\, f'(x)  -  f(x) \, g'(x)}{g(x)^2} \]

Example \(\PageIndex{2}\):

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)
                                                       
                          (x + 1)2

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

                         3
        =                               
                    (x + 1)2

   

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