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Katelyn!Kilmer!
MA!225!
11/3/2013!
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Proving!the!Quotient!Rule!for!Derivatives!
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When!computing!the!derivative!of!two!functions!that!are!divided!one!cannot!simply!
take!the!derivative!of!each!function!and!then!divided!them.!!Instead!the!quotient!rule!is!a!
formula!used!to!compute!these!values.!If!function!h(x)!is!the!result!of!two!functions!f(x)!and!g(x)!
being!divided,!the!quotient!rule!states:!
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proof.&
Let!f(x)!and!g(x)!be!functions!and!let!
By!the!limit!definition!of!a!derivative:!!
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This!is!a!complex!fraction!so!simplify!the!complex!fraction!by!getting!a!common!denominator!on!top:!
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Now!something!can!be!added!and!then!subtracting!so!essentially!zero!is!added!to!the!equation:!
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This!was!useful!because!now!the!functions!can!be!grouped!in!a!more!useful!way.!Functions!are!grouped!
and!separated:!
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Factor!out!a!one!from!the!second!equation:!
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Rewrite!equation!by!splitting!up!into!two!separate!limits!
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Factor!out!functions!that!can!be!factored!out:!
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Katelyn!Kilmer!
MA!225!
11/3/2013!
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A!g(x)!can!be!factored!out!of!the!first!limit:!
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Solve!the!limits!for!as!h!approaches!zero.!Since!the!limit!of!a!product!is!just!the!limit!of!each!function!
multiplied!together!each!piece!can!take!its!limit!and!then!multiplied!together.!!For!the!first!limit,!it!can!
be!split!into!two!products!where!one!is!the!limit!definition!of!a!derivative.!!For!the!second!limit,!the!
functions!f(x)!and!g(x)!depend!on!x.!x!is!a!variable!but!since!the!limit!only!depends!on!h!the!x!is!not!
effected!by!the!limit.!For!this!reason,!f(x)!and!g(x)!can!be!pulled!out!of!the!limit!
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This!equation!can!be!simplified!by!finding!a!common!denominator.!
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!!!!!!!!Therefore:!
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