bai giang dao ham rieng

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Bai giang dao ham rieng

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Page 1: Bai giang Dao ham rieng

Chapter 8: Partial Derivatives

Section 8.3Written by Dr. Julia Arnold

Associate Professor of MathematicsTidewater Community College, Norfolk Campus, Norfolk,

VAWith Assistance from a VCCS LearningWare Grant

Page 2: Bai giang Dao ham rieng

In this lesson you will learn•about partial derivatives of a function of two variables•about partial derivatives of a function of three or more variables•higher-order partial derivative

Page 3: Bai giang Dao ham rieng

Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.

Definition of Partial Derivatives of a Function of Two VariablesIf z = f(x,y), the the first partial derivatives of f with respect to x and y are the functions fx and fy defined by

0

0

, ( , ), lim

, ( , ), lim

x x

y y

f x x y f x yf x y

xf x y y f x y

f x yy

Provided the limits exist.

Page 4: Bai giang Dao ham rieng

To find the partial derivatives, hold one variable constant and differentiate with respect to the other.

Example 1: Find the partial derivatives fx and fy for the function4 2 2 3( , ) 5 2f x y x x y x y

Page 5: Bai giang Dao ham rieng

To find the partial derivatives, hold one variable constant and differentiate with respect to the other.

Example 1: Find the partial derivatives fx and fy for the function4 2 2 3( , ) 5 2f x y x x y x y

Solution: 4 2 2 3

3 2 2

2 3

( , ) 5 2

( , ) 20 2 6

( , ) 2 2x

y

f x y x x y x y

f x y x y x yx

f x y x y x

Page 6: Bai giang Dao ham rieng

Notation for First Partial DerivativeFor z = f(x,y), the partial derivatives fx and fy are denoted by

( , ) ,

( , ) ,

x x

y y

zf x y f x y zx xand

zf x y f x y zy y

The first partials evaluated at the point (a,b) are denoted by

( , ) ( , ), ,a b x a b yz zf a b and f a bx y

Page 7: Bai giang Dao ham rieng

Example 2: Find the partials fx and fy and evaluate them at the indicated point for the function

( , ) (2, 2)xyf x y atx y

Page 8: Bai giang Dao ham rieng

Example 2: Find the partials fx and fy and evaluate them at the indicated point for the function

( , ) (2, 2)xyf x y atx y

Solution:

2 2

2 2 2

2

2

2 2

2 2 2

2

2

( , ) (2, 2)

,( ) ( ) ( )

2 4 12, 216 4(2 2 )

,( ) ( ) ( )

4 12, 216 4( )

x

x

y

y

xyf x y atx yx y y xy xy y xy yf x yx y x y x y

f

x y x xy x xy xy xf x yx y x y x y

xfx y

Page 9: Bai giang Dao ham rieng

The following slide shows the geometric interpretation of the partial derivative. For a fixed x, z = f(x0,y) represents the curve formed by intersecting the surface z = f(x,y) with the plane x = x0.

0 0,xf x y represents the slope of this curve at the point (x0,y0,f(x0,y0))

Thanks to http://astro.temple.edu/~dhill001/partial-demo/For the animation.

Page 10: Bai giang Dao ham rieng
Page 11: Bai giang Dao ham rieng

Definition of Partial Derivatives of a Function of Three or More VariablesIf w = f(x,y,z), then there are three partial derivatives each of which is formed by holding two of the variables

0

0

0

, , ( , , ), , lim

, , ( , , ), , lim

, , ( , , ), , lim

x x

y y

z z

f x x y z f x y zw f x y zx x

f x y y z f x y zw f x y zy y

f x y z z f x y zw f x y zz z

In general, if

1 2

1 2

( , ,... )

, ,... , 1, 2,...k

n

x nk

w f x x x there are n partial derivativesw f x x x k nx

where all but the kth variable is held constant

Page 12: Bai giang Dao ham rieng

Notation for Higher Order Partial DerivativesBelow are the different 2nd order partial derivatives:

yx

xy

yy

xx

fyxf

yf

y

fxyf

xf

y

fyf

yf

y

fxf

xf

x

2

2

2

2

2

2Differentiate twice with respect to x

Differentiate twice with respect to y

Differentiate first with respect to x and then with respect to y

Differentiate first with respect to y and then with respect to x

Page 13: Bai giang Dao ham rieng

TheoremIf f is a function of x and y such that fxy and fyx are continuous on an open disk R, then, for every (x,y) in R, fxy(x,y)= fyx(x,y)

Example 3:Find all of the second partial derivatives of yxyxyyxf 22 523),(

Work the problem first then check.

Page 14: Bai giang Dao ham rieng

Example 3:Find all of the second partial derivatives of yxyxyyxf 22 523),(

xyyxf

xxyyxf

yxyxyyxf

xyyxfxyyyxf

yxyxyyxf

xyxf

xxyyxf

yxyxyyxf

yyxfxyyyxf

yxyxyyxf

yx

y

xy

x

yy

y

xx

x

106),(526),(

523),(

106),(103),(

523),(

6),(526),(

523),(

10),(103),(

523),(

2

22

2

22

2

22

2

22

Notice that fxy = fyx

Page 15: Bai giang Dao ham rieng

Example 4: Find the following partial derivatives for the function zxyezyxf x ln),,(

a. xzf

b. zxf

c. xzzf

d. zxzf

e. zzxf

Work it out then go to the next slide.

Page 16: Bai giang Dao ham rieng

Example 4: Find the following partial derivatives for the function zxyezyxf x ln),,(

a. xzf

b. zxf

zzyxf

zyezyxf

zxyezyxf

xz

xx

x

1),,(

ln),,(ln),,(

zzyxf

zxzyxf

zxyezyxf

zx

z

x

1),,(

),,(

ln),,(

Again, notice that the 2nd partials fxz = fzx

Page 17: Bai giang Dao ham rieng

c. xzzf

d. zxzf

e. zzxf

21),,(

1),,(

ln),,(ln),,(

zzyxf

zzyxf

zyezyxf

zxyezyxf

xzz

xz

xx

x

21),,(

1),,(

),,(

ln),,(

zzyxf

zzyxf

zxzyxf

zxyezyxf

zxz

zx

z

x

2

2

1),,(

),,(

),,(

ln),,(

zzyxf

zxzyxf

zxzyxf

zxyezyxf

zzx

zz

z

x

NoticeAll

Are Equal

Page 18: Bai giang Dao ham rieng

Go to BB for your exercises.