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What is the quotient rule

  1. Jul 23, 2014 #1

    The quotient rule is a formula for the derivative of the quotient of two functions, for which derivatives exist.


    [tex]f(x) = \frac{g(x)}{h(x)}[/tex]


    [tex]f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}[/tex]

    here, [itex]h(x) \: \neq \: 0[/itex]

    Extended explanation

    Even though a quotient can always be differentiated using the product and chain rules, it is easier and more efficient to remember and use the quotient rule. :wink:

    Proof of the quotient rule:
    [tex]f(x) = \frac{g(x)}{h(x)} = g(x)[h(x)]^{-1}[/tex]
    Using the product and chain rules:
    [tex]f'(x) = g'(x)\:[h(x)]^{-1} - \: [h(x)]^{-2} \: h'(x)\:g(x)[/tex]
    and, putting this over a common denominator:
    [tex]f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
  2. jcsd
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