Differentiating logarithm and exponential functions mcTYlogexp Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredierentiated fromrstprinciples
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Differentiating logarithm and exponential functions mcTYlogexp Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredierentiated fromrstprinciples

Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthistopicyoushouldbeableto di64256erentiate ln from64257rstprinciples di6425

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Differentiating logarithm and exponential functions mcTYlogexp Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredierentiated fromrstprinciples




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Presentation on theme: "Differentiating logarithm and exponential functions mcTYlogexp Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredierentiated fromrstprinciples"— Presentation transcript:


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Differentiating logarithm and exponential functions mc-TY-logexp-2009-1 Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredifferentiated fromfirstprinciples. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: differentiate ln fromfirstprinciples differentiate Contents 1. Introduction 2. Differentiationofafunction 3. Differentiationof ) = ln 4.

Differentiationof ) = e www.mathcentre.ac.uk 1 math centre2009
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1. Introduction Inthisunitweexplainhowtodifferentiatethefunctions ln and fromfirstprinciples. Tounderstandwhatfollowsweneedtousetheresultthatthee xponentialconstanteisdefined asthelimitas tendstozeroof (1 + /t i.e. lim (1 + /t Togetafeelforwhythisisso,wehaveevaluatedtheexpressi on (1 + /t foranumberof decreasingvaluesof asshowninTable1.Notethatas getsclosertozero,thevalueofthe expressiongetsclosertothevalueoftheexponentialconst ante 718 ... . Youshouldverify

someofthevaluesintheTable,andexplorewhathappensas reducesfurther. (1 + /t (1 + 1) =2 0.1 (1 + 0 1) =2.594 0.01 (1 + 0 01) 01 =2.705 0.001 (1 001) 001 =2.717 0.0001 (1 0001) 0001 =2.718 Wewillalsomakefrequentuseofthelawsofindicesandthela wsoflogarithms,whichshould berevisedifnecessary. 2. Differentiation of a function Recallthattodifferentiateanyfunction, ,fromfirstprincipleswefindtheslope, δy δx ,ofthe linejoininganarbitrarypoint, ,andaneighbouringpoint, ,onthegraphof . Wethen determinewhathappensto δy δx inthelimitas δx tendstozero.(SeeFigure1).

δx δx δy δx Figure1. δy δx istheslopeof AB Thederivative, ,isthengivenby ) = lim δx δy δx = lim δx δx δx Useofthisresulthasbeenexplainedatsomelengthinthefirs tunitondifferentiationfromfirst principles. www.mathcentre.ac.uk 2 math centre2009
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3. Differentiation of ) = ln Usingthedefinitionofthederivativeinthecasewhen ) = ln wefind δx δx ln( δx ln δx Weproceedbyusingthelawoflogarithms log log = log tore-writetheright-handside asfirstly δx (ln( δx ln ) =

δx ln δx δx ln 1 + δx Inordertosimplifywhatwillfollowwemakeasubstitution: let δx ,thatis, δx xt .(This substitutionismadebecauseinthecalculationswhichfoll owitistheratioof δx to which turnsouttobeimportant.Weneednotworryabout beingzerobecauseweareinterestedin differentiating ln andthelogarithmfunctionisonlydefinedforpositivevalue sof .) Then δx δx xt ln(1 + Further,usingthelaw log = log wecantakethe insidethelogarithmtogive δx δx ln(1 + ReferringtothegeneralcaseinFigure1,thisrepresentsth eslopeofthelinejoiningthetwo

pointsonthegraphof .Tofindthederivativeweneedtolet δx tendtozero.Becausewe substituted δx weneedtolet tendtozero. Wehave ) = lim ln (1 + Inthislimitingprocessitis whichtendstozero,andwecanregard asafixednumber. So, itcanbetakenoutsidethelimittogive: ) = lim ln (1 + Butweknowthat lim (1 + = e andso ) = ln e = since ln e = 1 Wehaveshown,fromfirstprinciples,thatthederivativeof ln isequalto www.mathcentre.ac.uk 3 math centre2009
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Key Point if ) = ln then ) = Exercise 1. Showfromfirstprinciples, usingexactlythesametechniq ue, thatif ) = log 10 then

) = ln10 2.Showfromfirstprinciplesthatif ) = log then ) = ln 4. Differentiation of ) = e Todifferentiate = e wewillrewritethisexpressioninitsalternativeformusin glogarithms: ln Thendifferentiatingbothsideswithrespectto (ln ) = 1 Theideaisnowtofind Recallthat (ln ) = (ln .(Thisresultisobtainedusingatechniqueknownas the chainrule .Youshouldrefertotheunitonthechainruleifnecessary). Nowweknow,fromSection3,that (ln ) = andso = 1 Rearranging, But = e andsowehavetheimportantandwell-knownresultthat = e Key Point if ) = e then ) = e www.mathcentre.ac.uk 4 math centre2009


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Theexponentialfunction(andmultiplesofit)istheonlyfu nctionwhichisequaltoitsderivative. Exercise 1.Showfromfirstprinciples,usingexactlythesametechniq ue,thatif ) = then ) = ln www.mathcentre.ac.uk 5 math centre2009