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[PDF] THE DERIVATIVE
Here, we represent the derivative of a function by a prime symbol For example, writing ? represents the derivative of the function evaluated at point
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Definition of Derivative • 6 Example • 7 Extension of the idea • 8 Example • 9 Derivative as a Function • 10 Rules of Differentiation
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For example, gold is widely considered a good investment to hedge against stocks and currencies When the stock market as a whole isn't performing well, or
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THEDERIVATIVE
Summary
1.Derivativeofusualfunctions...................................................................................................3
1.1.Constantfunction............................................................................................................3
1.2.Identityfunctionࢌሺ࢞ሻൌ ࢞............................................................................................3
1.3.Afunctionattheform࢞
................................................................................................31.4.Exponentialfunction(oftheformࢇ࢞withࢇ Ͳ):......................................................5
1.5.Functionࢋ࢞
......................................................................................................................5
1.6.Logarithmicfunction࢞...............................................................................................5
2.Basicderivationrules..............................................................................................................6
2.1.Multipleconstant............................................................................................................6
2.2.Additionandsubtractionoffunctions............................................................................6
2.3.Product
offunctionsrule.................................................................................................7
2.4.Quotientoffunctionsrule...............................................................................................8
3.Derivativeofcompositefunctions..........................................................................................9
Howdowerecognizeacompositefunction?.............................................................................9
3.1.Thechainrule..................................................................................................................9
3.2.Chainderivativesofusualfunctions..............................................................................10
4.Evaluationoftheslopeofthetangentatonepoint.............................................................12
5.Increasinganddecreasingfunctions.....................................................................................12
Theslopeconceptusuallypertainstostraightlines.Thedefinitionofastraightlineisa functionforwhichtheslopeisconstant.Inotherwords,nomatterwhichpointweare lookingat,theinclinationofalineremainsthesame.WhenafunctionisnonͲlinear,its slopemayvaryfromonepointtothenext.Wemustthereforeintroducethenotionof derivatewhichallowsustoobtaintheslopeatallpointsofthesenonͲlinearfunctions.Page2of13
Definition
Thederivativeofafunctionfatapointݔ,written݂ඁሺݔሻ,isgivenby: ݂ ඁ ሺݔሻൌο௫՜
݂ሺݔοݔሻെ ݂ሺݔሻοݔ
ifthislimitexists. Graphically,thederivativeofafunctioncorrespondstotheslopeofitstangentlineat onespecificpoint.Thefollowingillustrationallowsustovisualisethetangentline(in blue)ofagivenfunctionattwodistinctpoints.Notethattheslopeofthetangentline variesfromonepointtothenext.Thevalueofthederivativeofafunctiontherefore dependsonthepointinwhichwedecidetoevaluateit.Byabuseoflanguage,weoften speakoftheslopeofthefunctioninsteadoftheslopeofitstangentline.Notation
Here, werepresentthederivativeofafunctionbyaprimesymbol.Forexample,writing݂ඁሺݔሻrepresentsthederivativeofthefunction݂evaluatedatpointݔ.Similarly,writing
ሺ͵ݔ ʹሻඁindicateswearecarryingoutthederivativeofthefunction͵ݔ ʹ.Theprime
symboldisappearsassoonasthe derivativehasbeencalculated.