chapter 5 Implicit Differentiation隐函数微分

ImplicitDifferentiationObjective:Tofindderivativesoffunctionsthatwecannotsolvefory.Howtodoit?•Bynow,itshouldbeeasyforyoutotakethederivativeofanequationsuchasxxy735Ifyou'regivenanequationsuchas,youcanstillfigureoutthederivativebytakingthesquarerootofbothsides,whichgivesyouintermsof.Thisisknownasfindingthederivativeexplicitly.It'smessy,butpossiple.xxy7352yxImplicitDifferentiation•Itisnotnecessarytosolveanequationforyintermsofxinordertodifferentiatethefunctiondefinedimplicitlybytheequation(butoftenitiseasiertodoso).xyxy1Finddy/dxfor.Canwesolvethisfory?ImplicitDifferentiation•Itisnotnecessarytosolveanequationforyintermsofxinordertodifferentiatethefunctiondefinedimplicitlybytheequation.11xxy22)1(2)1()1)(1()1)(1(xxxxdxdyForexample,wecantakethederivativeofwiththequotientrule:ImplicitDifferentiation•Wecanalsotakethederivativeofthegivenfunctionwithoutsolvingforybyusingatechniquecalledimplicitdifferentiation.Wewilluseallofourpreviousrulesandstatetheindependentvariable.xyxy1Example1Findif•Usingimplicitdifferentiation,youget:dxdy452334xxyydxdxxdxdxxdxdyydxdyy34212583Rememberthat1dxdx34212583xxyydxdy）（Afteryoufactorout,dividebothsidesbydxdy:832yyyyxxdxdy83125234•Note:Nowthatyouunderstandthatthederivativeofantermwithrepecttowillalwaysbemultipliedby,andthat,wewon'twriteanymore.Youshouldunderstandthatthetermisimplied.xdxdx1dxdxdxdxxExample2Findif•Usingimplicitdifferentiation,youget:dxdy2222sincoscosxyxyThensimplify:1dxdxNext,putallofthetermscontainingontheleftandalloftheothertermsontheright:dxdyxxdxdyyyxxdxdyyy2cos2sin2sin2cos22222222cos2sin2sin2cos2xxdxdyyyxxdxdyyy2222cos2sin2sin2cos2xxxxdxdyyydxdyyyNext,factoroutdxdy2222cos2sin2sin2cos2xxxxdxdyyyyy）（Andisolate:dxdy）（2222sin2cos2cos2sin2yyyyxxxxdxdyThiscanbesimplifiedfurtherto:）（2222sincos)cos(sinyyyxxxdxdyExample3FindifImplicitdifferentiationshouldresultin:dxdyYoucansimplifythisto:Next,putallofthetermscontainingontheleftandalloftheothertermsontheright:dxdyNext,factoroutdxdy8453322yxyx012525622dxdyyydxdyyxx012510622dxdyyydxdyxyx22561210yxdxdyydxdyxyAndisolate:dxdy22561210yxdxdyyxy）（）（22121056yxyyxdxdyExample4:Findthederivativeofat(2,1).94322yyxYouneedtouseimplictdifferentiationtofind:dxdy086dxdydxdyyxNow,insteadofrearrangingtoisolate,plugin(2,1)immediatelyandsolveforthederivative;dxdy01826dxdydxdySimplify:07-12dxdyso712dxdyExample5:Findthederivativeofat(1,1).First,cross-multiply:Takethederivative:xxyyx232452232452xyxyxDistribute:3324-52xxyyx2323434-102xydxdyyxdxdyyDonotsimplifynow.Rather,plugin(1,1)rightaway.Thiswillsaveyoufromthealgebra:23213141314-1102dxdydxdyNowsolvefor:dxdy112-102dxdydxdy3-2dxdy23-dxdySecondderivativeExample6:Findifxdyd22xxyy24222Differentiatingimplicity,youget:2822xdxdydxdyyNext,simplifyandsolvefor:dxdy114yxdxdyNowit'stimetotakethederivativeagain:22211414ydxdyxyxdydFinally,substitutefordxdy211141414yyxxy22211414yxyProblem1.Findif.dxdyxyyx622Problem2.Findif.dxdyxyyxcosProblem3.Findthederivativeofeachvariablewithrespecttooft222zyxProblem4.FindthederivativeofeachvariablewithrespecttoofthrV231Problem5:Findifxdyd22xxy222Example1•Useimplicitdifferentiationtofinddy/dxif22sin5xyyExample2•Useimplicitdifferentiationtofinddy/dxif22sin5xyyxdxdyydxdyy2cos10Example2•Useimplicitdifferentiationtofinddy/dxif22sin5xyyxdxdyydxdyy2cos10xdxdyyy2)cos10(Example2•Useimplicitdifferentiationtofinddy/dxif22sin5xyyxdxdyydxdyy2cos10xdxdyyy2)cos10(yyxdxdycos102Example3•Useimplicitdifferentiationtofindif22dxyd.92422yxExample3•Useimplicitdifferentiationtofindif22dxyd.92422yx048dxdyyxExample3•Useimplicitdifferentiationtofindif22dxyd.92422yx048dxdyyxyxdxdy2Example3•Useimplicitdifferentiationtofindif22dxyd.92422yxyxdxdy22222)2(ydxdyxydxydExample3•Useimplicitdifferentiationtofindif22dxyd.92422yxyxdxdy22222)2(ydxdyxydxyd222222yyxxydxydExample3•Useimplicitdifferentiationtofindif22dxyd.92422yxyxdxdy22222)2(ydxdyxydxyd222222yyxxydxyd3222242yxydxydExample3•Useimplicitdifferentiationtofindif22dxyd.92422yxyxdxdy22222)2(ydxdyxydxyd222222yyxxydxyd3222242yxydxyd3229ydxydExample4•Findtheslopesofthetangentlinestothecurveatthepoints(2,-1)and(2,1).012xyExample4•Findtheslopesofthetangentlinestothecurveatthepoints(2,-1)and(2,1).•Weknowthattheslopeofthetangentlinemeansthevalueofthederivativeatthegivenpoints.012xyExample4•Findtheslopesofthetangentlinestothecurveatthepoints(2,-1)and(2,1).•Weknowthattheslopeofthetangentlinemeansthevalueofthederivativeatthegivenpoints.012xy012dxdyyydxdy21Example4•Findtheslopesofthetangentlinestothecurveatthepoints(2,-1)and(2,1).•Weknowthattheslopeofthetangentlinemeansthevalueofthederivativeatthegivenpoints.012xy012dxdyyydxdy212112yxdxdy2112yxdxdyExample5Us