Afterreadingthistextandorviewingthevideotutorialo nthistopicyoushouldbeableto di64256erentiatethefunction sin from64257rstprinciples di64256erentiatethefunction cos from64257rstprinciples Contents 1 Introduction 2 Thederivativeof sin 3 Thederivati ID: 23726
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Differentiationofthesineandcosinefunctionsfromrstprinciples mc-TY-sincos-2009-1 Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpracticeexercisessothattheybecomesecondnature.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:di erentiatethefunctionsinxfrom rstprinciplesdi erentiatethefunctioncosxfrom rstprinciplesContents1.Introduction22.Thederivativeoff(x)=sinx33.Thederivativeoff(x)=cosx4 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionInthisunitwelookathowtodi erentiatethefunctionsf(x)=sinxandf(x)=cosxfrom rstprinciples.Weneedtoremindourselvesofsomefamiliarresults.Thederivativeoff(x).Thede nitionofthederivativeofafunctiony=f(x)isdy dx=limx!0f(x+x) f(x) xTwotrigonometricidentities.WewillmakeuseofthetrigonometricidentitiessinC sin=2cosC+ 2sinC 2cosC cos= 2sinC+ 2sinC 2Thelimitofthefunctionsin .As(measuredinradians)approacheszero,thefunctionsin tendsto1.Wewritethisaslim!0sin =1Thisresultcanbejusti edbychoosingvaluesofcloserandclosertozeroandexaminingthebehaviourofsin .Table1showsvaluesofandsin asbecomessmaller.sinsin 10.841470.841470.10.099830.998330.010.009990.99983 Table1:Thevalueofsin astendstozerois1.Youshouldverifytheseresultswithyourcalculatortoappreciatethatthevalueofsin ap-proaches1astendstozero.Wenowusetheseresultsinordertodi erentiatef(x)=sinxfrom rstprinciples. www.mathcentre.ac.uk2c\rmathcentre2009 2.Differentiatingf(x)=sinxHeref(x)=sinxsothatf(x+x)=sin(x+x).Sof(x+x) f(x)=sin(x+x) sinxTherighthandsideisthedi erenceoftwosineterms.Weusethe rsttrigonometricidentity(above)towritethisinanalternativeform.sin(x+x) sinx=2cosx+x+x 2sinx 2=2cos2x+x 2sinx 2=2cos(x+x 2)sinx 2Then,usingthede nitionofthederivativedy dx=limx!0f(x+x) f(x) x=2cos(x+x 2)sinx 2 xThefactorof2canbemovedintothedenominatorasfollows,inordertowritethisinanalternativeform:dy dx=cos(x+x 2)sinx 2 x=2=cosx+x 2sinx 2 x 2Wenowletxtendtozero.Considerthetermsinx 2 x 2andusetheresultthatlim!0sin =1with=x 2.Weseethatlimx!0sinx 2 x 2=1Further,limx!0cosx+x 2=cosxSo nally,dy dx=cosx www.mathcentre.ac.uk3c\rmathcentre2009 3.Thederivativeoff(x)=cosx.Heref(x)=cosxsothatf(x+x)=cos(x+x).Sof(x+x) f(x)=cos(x+x) cosxTherighthandsideisthedi erenceoftwocosineterms.ThistimeweusethetrigonometricidentitycosC cos= 2sinC+ 2sinC 2towritethisinanalternativeform.cos(x+x) cosx= 2sinx+x+x 2sinx 2 2sin2x+x 2sinx 2= 2sin(x+x 2)sinx 2Then,usingthede nitionofthederivativedy dx=limx!0f(x+x) f(x) x= 2sin(x+x 2)sinx 2 xThefactorof2canbemovedasbefore,inordertowritethisinanalternativeform:dy dx= sin(x+x 2)sinx 2 x=2= sinx+x 2sinx 2 x 2Wenowwanttoletxtendtozero.Asbeforelimx!0sinx 2 x 2=1Further,limx!0 sinx+x 2= sinxSo nally,dy dx= sinxSo,wehaveuseddi erentiationfrom rstprinciplesto ndthederivativesofthefunctionssinxandcosx. www.mathcentre.ac.uk4c\rmathcentre2009