2DerivativeandRulesofDifferentiationWord文件下载.docx
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▪NoteC=15,i.e.
Cisstill(mathematically)afunctionofquantitybecauseitsatisfiedthedefinition
15,15,15,15,...
Derivatives
Acommonproblemisthis:
IfC=f(Q),HowdoesCchangewhenQchanges?
i.e.WhatistherateofchangeofCwithregardto(WRT)Q?
TheanswerisgivenbythederivativeofCWRTQ.
Thederivativeiswritten
orC’
Ineconomicsthederivativeisequivalenttoamarginalquantity(Marginalcost,MarginalRevenue,MarginalProduct)
IngeneralY=f(X)
Where:
Yisthedependentvariable
Xistheindependentvariable
Derivative→
orY`
Itis
(a)therateofchangeofYWRTX
or
(b)byhowmuchdoesYchangewhenXchangesbyone(verysmall)unit?
(c)onagraphtheslope(sameas(b))
Slopeis
=
RulesforDerivatives
PowerFunctionRule
D1IfY=aXb[a,b=constants,numbers]
Y´
=baXb-1
e.g.
C=3Q2
C´
=2*3Q2-1
=6Q1
=6Q
Therulecanbeappliedtofunctionwithseveralterms.
e.g.
C=10+4Q3+6Q
=12Q2+3Q
Derivativeofaconstant(10)=0
C=10+3Q
=3Q1-1
=3Q0
=3
ProductRule
SayC=(3+2Q)(1+Q2)C’=?
D2IfUandVarefunctionsofX
C=(3+2Q)(1+Q2)
=(3+2Q)2Q+(1+Q2)2
=6Q+4Q2+2+2Q2
=2+6Q+6Q2
QuotientRule
D3IfUandVarefunctionsofX
C=
C´
=2
ChainRule
SayC=3+10Q
Q=20-2P
P→Q→C:
changeP→changeinQ→changeinC
Whatis
?
Howcanwefind
D4X=f(Y)
Y=f(Z)
C=3+10Q
Q=20-2P
=(10)*(-2)
=-20
Y=
=?
Y=
Y=f(Z)
Z=F(X)
=
Longchains:
SayW=f(Y)
Y=g(X)
X=h(Z)
etc.
InverseRule
Sometimeswehave
butwewant
.Theruleissimple
D5
(i)C=10+3Q
=?
=3
=
(ii)C=10+3Q+0.1Q2
=3+0.2Q
ExponentialFunction
▪eisanumber=2.71828…
[
]
▪ConsidereX.XisthePower(ThisisnotlikeXa)
D6
(ThederivativehasthesamevalueaseX)
Illustration:
X=
1
2
3
eX=
2.7
7.4
20.1
(Note:
thesefiguresaretoonedecimalplace)
i.e.WhenX=2;
ex=2.7andonesmallunitchangeinXwillcauseextoincreaseby2.7ofthoseunits.SoasXgetsbigger,eXincreasesfaster.Itspeedsup.
ConsidereaX(a=aconstantornumber)
D7
(Notthesameas
)
e.g.
▪NowconsiderlogeX
(alsowrittenlogXorlnXorthenaturallogofX)
D8
e.g.Y=3logX
TherateofchangeoflogXgetssmallerasXgetsbigger(Itslowsdown)
Wecanprovethisifweassume
X=eY
LogX=Y
Y=logX
Then
[D5]