2DerivativeandRulesofDifferentiation.docx

2DerivativeandRulesofDifferentiation.docx

《2DerivativeandRulesofDifferentiation.docx》由会员分享，可在线阅读，更多相关《2DerivativeandRulesofDifferentiation.docx（9页珍藏版）》请在冰豆网上搜索。

2DerivativeandRulesofDifferentiation

SimpleDerivatives

EconomicTheory→Relationshipsbetweenvariables→functions

e.g.Pricedependsonquantitysold

Costdependsonquantitysold

Employmentdependsoninvestment

Costisafunctionofquantity

（Foreveryvalueofquantitythereexistsoneandonlyonevalueofcost）→C=f（Q）,C=（Q）orevenC=C（Q）

▪Specifically:

Maybewrittenas

C=10+3Q

Q=1,2,3,4,...

13

16,19,22,...

▪NoteC=15,i.e.

Cisstill（mathematically）afunctionofquantitybecauseitsatisfiedthedefinition

Q=1,2,3,4,...

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?

or

（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

C´=

=12Q2+3Q

Derivativeofaconstant（10）=0

C=10+3Q

C´=3Q1-1

=3Q0

=3

ProductRule

SayC=（3+2Q）（1+Q2）C’=?

D2IfUandVarefunctionsofX

e.g.

C=（3+2Q）（1+Q2）

C´=（3+2Q）2Q+（1+Q2）2

=6Q+4Q2+2+2Q2

=2+6Q+6Q2

QuotientRule

D3IfUandVarefunctionsofX

e.g.

C=

C´=

=2

ChainRule

SayC=3+10Q

Q=20-2P

P→Q→C:

changeP→changeinQ→changeinC

Whatis

?

Howcanwefind

?

D4X=f（Y）

Y=f（Z）

e.g.

C=3+10Q

Q=20-2P

C´=

=（10）*（-2）

=-20

e.g.

Y=

Y´=?

Y=

Y=f（Z）

Z=F（X）

=

=

Longchains:

SayW=f（Y）

Y=g（X）

X=h（Z）

etc.

InverseRule

Sometimeswehave

butwewant

.Theruleissimple

D5

e.g.

（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=

0

1

2

3

eX=

1

2.7

7.4

20.1

=

1

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]