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# Differentiating logarithm and exponential functions mcTYlogexp Thisunitgivesdetailsofhowlogarithmicfunctionsandexp onentialfunctionsaredierentiated fromrstprinciples

Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthistopicyoushouldbeableto di64256erentiate ln from64257rstprinciples di6425

## 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