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ImplicitDifferentiation Objective Tofindderivativesoffunctionsthatwecannotsolvefory Howtodoit Bynow itshouldbeeasyforyoutotakethederivativeofanequationsuchas Ifyou regivenanequationsuchas youcanstillfigureoutthederivativebytakingthesquarerootofbothsides whichgivesyouintermsof Thisisknownasfindingthederivativeexplicitly It smessy butpossiple ImplicitDifferentiation Itisnotnecessarytosolveanequationforyintermsofxinordertodifferentiatethefunctiondefinedimplicitlybytheequation butoftenitiseasiertodoso Finddy dxfor Canwesolvethisfory ImplicitDifferentiation Itisnotnecessarytosolveanequationforyintermsofxinordertodifferentiatethefunctiondefinedimplicitlybytheequation Forexample wecantakethederivativeofwiththequotientrule ImplicitDifferentiation Wecanalsotakethederivativeofthegivenfunctionwithoutsolvingforybyusingatechniquecalledimplicitdifferentiation Wewilluseallofourpreviousrulesandstatetheindependentvariable Example1Findif Usingimplicitdifferentiation youget Rememberthat Afteryoufactorout dividebothsidesby Note Nowthatyouunderstandthatthederivativeofantermwithrepecttowillalwaysbemultipliedby andthat wewon twriteanymore Youshouldunderstandthatthetermisimplied Example2Findif Usingimplicitdifferentiation youget Thensimplify Next putallofthetermscontainingontheleftandalloftheothertermsontheright Next factorout Andisolate Thiscanbesimplifiedfurtherto Example3Findif Implicitdifferentiationshouldresultin Youcansimplifythisto Next putallofthetermscontainingontheleftandalloftheothertermsontheright Next factorout Andisolate Example4 Findthederivativeofat 2 1 Youneedtouseimplictdifferentiationtofind Now insteadofrearrangingtoisolate plugin 2 1 immediatelyandsolveforthederivative Simplify so Example5 Findthederivativeofat 1 1 First cross multiply Takethederivative Distribute Donotsimplifynow Rather plugin 1 1 rightaway Thiswillsaveyoufromthealgebra Nowsolvefor Secondderivative Example6 Findif Differentiatingimplicity youget Next simplifyandsolvefor Nowit stimetotakethederivativeagain Finally substitutefor Problem1 Findif Problem2 Findif Problem3 Findthederivativeofeachvariablewithrespecttoof Problem4 Findthederivativeofeachvariablewithrespecttoof Problem5 Findif Example1 Useimplicitdifferentiationtofinddy dxif Example2 Useimplicitdifferentiationtofinddy dxif Example2 Useimplicitdifferentiationtofinddy dxif Example2 Useimplicitdifferentiationtofinddy dxif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example3 Useimplicitdifferentiationtofindif Example4 Findtheslopesofthetangentlinestothecurveatthepoints 2 1 and 2 1 Example4 Findtheslopesofthetangentlinestothecurveatthepoints 2 1 and 2 1 Weknowthattheslopeofthetangentlinemeansthevalueofthederivativeatthegivenpoints Example4 Findtheslopesofthetangentlinestothecurveatthepoints 2 1 and 2 1 Weknowthattheslopeofthetangentlinemeansthevalueofthederivativeatthegivenpoints Example4 Findtheslopesofthetangentlinestothecurve
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