Differentiation by taking logarithms mcTYditakelogs Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewedier entiatethem
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Differentiation by taking logarithms mcTYditakelogs Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewedier entiatethem

Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthistopicyoushouldbeableto uselogarithmstosimplifyfunctionsbeforedi64256eren

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Differentiation by taking logarithms mcTYditakelogs Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewedier entiatethem




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Presentation on theme: "Differentiation by taking logarithms mcTYditakelogs Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewedier entiatethem"— Presentation transcript:


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Differentiation by taking logarithms mc-TY-difftakelogs-2009-1 Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewediffer- entiatethem. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: uselogarithmstosimplifyfunctionsbeforedifferentiatio Contents 1. Introduction 2. Someexamples www.mathcentre.ac.uk 1 math centre2009
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1. Introduction

Inthisunitwelookathowwecanuselogarithmstosimplifyce rtainfunctionsbeforewediffer- entiatethem.Tostartoff,weremindyouaboutlogarithmsthe mselves. If = ln ,thenaturallogarithmfunction,orthelogtothebaseeof ,then dy dx Youshouldbefamiliaralreadywiththisresult.Ifnot,yous houldlearnit,andyoucanalsorefer totheunitondifferentiationofthelogarithmandexponenti alfunctions. Wewillalsoneedtodifferentiate = ln ,where issomefunctionof .Inthiscase, dy dx Youshouldlearnthisresulttoo3 Key Point if = ln then dy dx if = ln then dy dx Wewillalsomakeuseofthefollowinglawsoflogarithms: Key

Point Lawsoflogarithms: log + log = log AB, log log = log , m log = log Theselawscanbeusedtosimplifylogarithmsofanybase,but thesamebasemustbeused throughoutaparticularcalculation. www.mathcentre.ac.uk 2 math centre2009
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2. Some examples Example Supposewewanttodifferentiate = ln(3 + 7) Usingthelawsoflogarithmswecanwrite as = 5 ln(3 + 7) Thisismorestraightforwardtodifferentiatebecausewehav = ln with ) = 3 + 7 TherulegivenintheKeyPointonpage2tellsusthat dy dx = 5 12 + 7 60 + 7 Whatstartedoutasquiteacomplicatedproblemwassimplifie dusingthelawsoflogarithms

beforewecarriedoutthedifferentiation. Example Supposewewanttodifferentiate = ln 1 + 2 Inthisexamplewehavethelogofaquotient.Wecanusethelaw ln = ln ln torewrite thisas = ln(1 ln(1 + 2 Thetwofunctionsontherightareeasytodifferentiateusing theKeyPointonpage2: dy dx 1 + 2 Wecanwritetheanswerasasingletermbywritingthembothov eracommondenominator: dy dx 3(1 + 2 2(1 (1 )(1 + 2 2 + 6 (1 )(1 + 2 (1 )(1 + 2 Example Supposewewanttodifferentiate sin Theproblemhereisthatthefunction sin appearsasapower. Bytakinglogarithmsofboth sides,andusingthelawsoflogarithmswecanavoidthisasfo llows:

Takinglogsofbothsidesgives: ln = ln sin andusingthethirdofthelawsgivenonpage2: ln = sin ln Thetermontherightisaproductof sin and ln andwewillbeabletousetheproductrule todifferentiateit. www.mathcentre.ac.uk 3 math centre2009
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Thetermontheleft,being ln ,needstobedifferentiatedimplicitly.Recallthat dx (ln ) = dy (ln dy dx dy dx So, dy dx = sin + ln cos sin ln cos sothat dy dx sin ln cos sin sin ln cos sincewerecallthat sin Example Supposewewishtodifferentiate (1 1 + Weproceedbytakinglogarithmsofbothsidestogive: ln = ln (1 1 +

Wethenusethelawsoflogarithmstorewritetherighthandsi deas ln = ln(1 ln (1 + Withafurtherapplicationofthelawsoflogarithms,andnot ingthat 1 + = (1 + ,we obtain ln = 3 ln(1 ln(1 + Differentiating,andrememberingtodifferentiatethelefth andsideimplicitly dy dx = 3 1 + 1 + 6(1 + (1 (1 )(1 + + 2 (1 )(1 + (1 )(1 + sothat dy dx (1 )(1 + (1 1 + (1 )(1 + www.mathcentre.ac.uk 4 math centre2009
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Thiscanbesimpifiedto dy dx (6 + + 4 )(1 (1 + Fromtheseexampleswehaveseenhowusinglogarithmstowrit ecertainfunctionsinalternative formscanhelpwhenweneedtodifferentiatethem.

Exercises 1.Findthederivativeofeachofthefollowingfunctions a) ln( + 1) b) ln(sin c) ln(( + 2 + 1) d) ln + 1 e) ln( sin 2.Bytakinglogsandusingimplicitdifferentiation,findthe derivativesofthefollowingfunctions a) 1 + 2 b) c) ln d) (1 + (1 e) ax Answers 1.a) + 1 b) cot c) 4(3 + 2) + 2 + 1 d) + 1 + 1)( 1) e) + cot 2.a) 1 + 2 + 3 (1 + 2 b) (ln + 1) c) ln ln d) (1 + )(1 + (1 e) ax ax (ln + 1) + www.mathcentre.ac.uk 5 math centre2009