chapter 7 : partial differentiation....determine the first partial derivative of z w.r.t y:.. treat...

CHAPTER 7 : PARTIAL DIFFERENTIATION.

© John Wiley and Sons 2013 www.wiley.com/college/Bradley © John Wiley and Sons 2013

Essential Mathematics for Economics and Business, 4th Edition

http://www.wiley.com/college/Bradley

www.wiley.com/college/Bradley © John Wiley and Sons 2013

• What is partial differentiation • How to get first order partial derivatives • How to get second order partial derivatives • Worked Example 7.2 • Worked Example 7.3 • Worked Example 7.4



Graphs and derivatives for function of one variable

Ordinary differentiation:

Example

26249)( 23 +−+−= xxxxf 26249 23 +−+−= xxxy

Graph: a curve in 2 dimensions

OR

24183 2 −+−= xxdxdy

18622

+−= xdx

yd



Functions of several variables

Copyright© 2013 Teresa Bradley and John Wiley & Sons Ltd

..treat y as a constant

Example

Graph: a surface in 3 dimensions

OR 42),( ++= yxyxf 42 ++= yxz

To differentiate z partially with respect to x

Partial differentiation

..then differentiate w.r.t. x:

xzxz or ∂∂

N.B. ∂ ..denotes partial differentiation d ..denotes ordinary differentiation

..denote the partial derivative of z wrt x

--see next slide



Determine the first partial derivative of z w.r.t x:

Worked Example 7.2(a) 42 ++= yxz


..then differentiate z w.r.t. x:

4][2 ++= yxz

1001 =++=∂∂xz

..since derivatives of constant terms (2y and 4) are zero

xzxz or ∂∂



Determine the first partial derivative of z w.r.t y: or zy

Worked Example 7.2(b) 42 ++= yxz

..treat x as a constant

..then differentiate w.r.t. y:

yz∂∂

42][ ++= yxz

20)1(20 =++=∂∂yz

..since derivatives of constant terms (x and 4) are zero



Worked Example 7.3

Find the first-order partial derivatives for each of the following functions.

(a) (b) (c)

532 2 ++= yxxz

3.07.010 KLQ =

U x y= 2 5



Determine the first partial derivative of z w.r.t x:


..then differentiate w.r.t. x: yxxz )1(3)2(2 +=∂∂

532 2 ++= yxxzWorked Example 7.3(a) 5][32 2 ++= yxxz

yxxz 34 +=∂∂

xzxz or ∂∂



Determine the first partial derivative of z w.r.t y:


..then differentiate z w.r.t. y:

yzyz or ∂∂

0)1(30 ++=∂∂ xyz

532 2 ++= yxxzWorked Example 7.3(a) 5][3][2 2 ++= yxxz

xyz 3=∂∂



Determine the first partial derivative of Q wrt L:

..treat K as a constant

..then differentiate Q w.r.t. L:

LQLQ or ∂∂

Worked Example 7.3(b) 3.07.010 KLQ =

][10 3.07.0 KLQ =

( ) 3.017.07.010 KLLQ −=∂∂

3.03.07 KLLQ −=∂∂



the first partial derivative of Q wrt K:

..treat L as a constant

..then differentiate Q w.r.t. K:

KQKQ or ∂∂


3.07.0 ][10 KLQ =

( )13.07.0 3.010 −=∂∂ KLKQ

7.07.03 −=∂∂ KLKQ



the first partial derivative of U wrt x:


..then differentiate U w.r.t. x:

xU∂∂

Worked Example 7.3(c)

( ) 52 yxxU

=∂∂

U x y= 2 5

][ 52 yxU =

52xyxU

=∂∂



the first partial derivative of U wrt y:


..then differentiate U w.r.t. y:

yU∂∂


)5( 42 yxyU

=∂∂

U x y= 2 5

52][ yxU =

425 yxyU

=∂∂



Worked Example 7.4

Find the second order partial derivatives for each of the following functions.

(a)

(b)

(c)

532 2 ++= yxxz

3.07.010 KLQ =

U x y= 2 5



or zxx : second partial derivative of z wrt x:


..differentiate

2

2

xz

∂∂

532 2 ++= yxxzWorked Example 7.4(a)

yxxz 34 +=∂∂

..first partial derivative zx see Worked Example 7.3

][34 yxxz

+=∂∂

04)1(422

+==∂∂xz

xz∂∂ w.r.t x



the second partial derivative of z wrt y:


..differentiate w.r.t. y:

2

2

yz

∂∂


xyz 3=∂∂

..first partial derivative zy see Worked Example 7.3

][3 xyz=

∂∂

022

=∂∂yz

yz∂∂



the second partial derivative of z wrt x and y:


..differentiate w.r.t. y:

xyz∂∂

∂2


yxxz 34 +=∂∂

..first partial derivative zx see Worked Example 7.3

xz∂∂

yxxz 3][4 +=∂∂

3)1(302

=+=∂∂

∂xy

z



or QLL: second partial derivative of Q wrt L

..treat K as a constant

..then differentiate QL w.r.t. L:

2

2

LQ

∂∂


( ) 3.013.022

3.07 KLLQ −−−=

∂

∂

3.03.07 KLLQ −=∂∂

..first partial derivative: QL

][7 3.03.0 KLLQ −=∂∂

3.03.12

21.2 KL

LQ −−=

∂

∂



: the second partial derivative of Q wrt K



..differentiate QK w.r.t. K:

2

2

KQ

∂∂


( )17.07.022

7.03 −−−=∂

∂ KLK

Q

7.07.03 −=∂∂ KLKQ

..first partial derivative QK

7.17.02

21.2 −−=

∂

∂ KLK

Q

7.07.0 ][3 −=∂∂ KLKQ



the second ‘mixed’ partial derivative of Q



..differentiate QL w.r.t. K:

LKQ∂∂

∂2


( )13.03.02 3.07 −−=∂∂

∂ KLLK

Q

3.03.07 KLLQ −=∂∂..first partial derivative: QL

3.03.0 ][7 KLLQ −=∂∂

7.03.12

1.2 −−=∂∂

∂ KLLK

Q



the second partial derivative: Uxx



..then differentiate Ux w.r.t. x:

2

2

xU

∂∂


][2 5yxxU

=∂∂

U x y= 2 5

52xyxU

=∂∂

..first partial derivative: Ux

552

22)1(2 yy

xU

==∂

∂



the second partial derivative: Uyy



..then differentiate Uy w.r.t. y:

2

2

yU

∂∂

Worked Example 7.4(c) U x y= 2 5

425 yxyU

=∂∂

..first partial derivative: Uy

32322

220)4(5 yxyx

yU

==∂

∂

42][5 yxyU

=∂∂



the second mixed derivative: Uxy


..then differentiate Ux w.r.t. y:

xyU∂∂

∂2


5][2 yxxU

=∂∂

U x y= 2 5

52xyxU

=∂∂

..first partial derivative: Ux

442

10)5(2 xyyxxy

U==

∂∂∂


Chapter 7 : partial differentiation.�Slide Number 2Graphs and derivatives for function of one variableFunctions of several variables Determine the first partial derivative of z w.r.t x: �Determine the first partial derivative of z w.r.t y: or zy �Worked Example 7.3Determine the first partial derivative of z w.r.t x:Determine the first partial derivative of z w.r.t y: �Determine the first partial derivative of Q wrt L: the first partial derivative of Q wrt K: the first partial derivative of U wrt x: the first partial derivative of U wrt y: Worked Example 7.4 or zxx : second partial derivative of z wrt x: the second partial derivative of z wrt y: the second partial derivative of z wrt x and y: or QLL: second partial derivative of Q wrt L : the second partial derivative of Q wrt K the second ‘mixed’ partial derivative of Q the second partial derivative: Uxx the second partial derivative: Uyy the second mixed derivative: Uxy

chapter 7 : partial differentiation....determine the first partial derivative of z w.r.t y:.. treat...

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