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Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis", 3rd ed., Prentice Hall. Fundamental Methods of Mathematical Economics, Kevin Wainwright, Alpha Chiang. Mathematics for Economists, Carl P. Simon, Lawrence E. Blume

Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

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Page 1: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Readings for those who have problems with calculus.

Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis", 3rd ed., Prentice Hall.

Fundamental Methods of Mathematical Economics, Kevin Wainwright, Alpha Chiang.

Mathematics for Economists, Carl P. Simon, Lawrence E. Blume

Page 2: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Derivatives - introduction

• Suppose a car is accelerating from 30mph to 50mph.

• At some point it hits the speed of 40mph, but when?

• Speed = (distance travelled)/(time passed)• How is it possible to define the speed at a

single point of time?

Page 3: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Approximations

• Idea: find distance/time for smaller and smaller time intervals.

Here f shows how the distance is changing with the time, x.

The time difference is h while the distance travelled is

f(x+h) – f(x).

Page 4: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

The derivative

• Our approximation of the speed is

• As h gets smaller, the approximation of the speed gets better.

• When h is infinitesimally small, the calculation of the speed is exact.

• Leibniz used the notation dy/dx for the exact speed.

Page 5: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Example

• f(x) = x2

• f(x+h) – f(x) = (x + h)2 – x2

= (x2 + 2xh + h2) - x2 = 2xh + h2

• Speed = (2xh + h2)/h = 2x + h• When h is infinitesimally small, this is just 2x.

Page 6: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

0dc

dx

example: 3y

0y

The derivative of a constant is zero.

Rules for Differentiation

Page 7: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

We saw that if , .2y x 2y x

This is part of a pattern.

1n ndx nx

dx

examples:

4f x x

34f x x

8y x

78y x

power rule

Rules for Differentiation

Page 8: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

1n ndx nx

dx

Rules for Differentiation

Proof:

h

xhxx

dx

d nn

h

n

)(lim

0

h

xhhnxxx

dx

d nnnn

h

n

...lim

1

0

h

hhnxx

dx

d nn

h

n

...lim

1

0

1

0lim

n

h

n nxxdx

d

Page 9: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

d ducu c

dx dx

examples:

1n ndcx cnx

dx

constant multiple rule:

5 4 47 7 5 35dx x x

dx

Rules for Differentiation

Page 10: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

(Each term is treated separately)

d ducu c

dx dxconstant multiple rule:

sum and difference rules:

d du dvu v

dx dx dx d du dv

u vdx dx dx

4 12y x x 34 12y x

4 22 2y x x

34 4dy

x xdx

Rules for Differentiation

Page 11: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

product rule:

d dv duuv u v

dx dx dx Notice that this is not just the

product of two derivatives.

This is sometimes memorized as: d uv u dv v du

2 33 2 5d

x x xdx

2 3x 26 5x 32 5x x 2x

Rules for Differentiation

Page 12: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

quotient rule:

2

du dvv ud u dx dx

dx v v

or 2

u v du u dvdv v

3

2

2 5

3

d x x

dx x

2 2 3

22

3 6 5 2 5 2

3

x x x x x

x

Rules for Differentiation

Page 13: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Formulas you should learn

(Cxa)’=Cax(a-1); C, a – a real number(ex)’=ex

(ax)’=axlna; a>0

0,1

)'(ln xx

x

0,1,0,ln

1)'(log xaa

axxa

Page 14: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Derivatives rules - summary

, xgxxgx ff

2

fff

xg

xgxxgx

xg

x

0xg

xcxc ff

,

,,

for

c is a constant

xgxxgxxgx fff

0C ,

Page 15: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Consider a simple composite function:6 10y x

2 3 5y x

If 3 5u x

then 2y u

6 10y x 2y u 3 5u x

6dy

dx 2

dy

du 3

du

dx

dy dy du

dx du dx

6 2 3

Chain Rule

Page 16: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

dy dy du

dx du dx Chain Rule:

example: sinf x x 2 4g x x Find: at 2f g x

cosf x x 2g x x 2 4 4 0g

0 2f g cos 0 2 2 1 4 4

Chain Rule

If is the composite of and , then:f g y f u u g x

at at xu g xf g f g )('))((' xgxgf

Page 17: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Differentiation of Multivariate Functions

• The partial derivative of a multivariate function f(x,y) with respect to x is defined as

h

yxfyhxfxyxf

h

,,lim

,0

Page 18: Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis",

Differentiation of Multivariate Functions

f(x1,x2)= Cx1ax2

b

)(*),( )1(

12

1

21

ab axCxx

xxf

)(*),( )1(

21

2

21

ba bxCxx

xxf