Bai giang Dao ham rieng
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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
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
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.
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
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
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
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
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
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.
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
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
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.
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
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.
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
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
Go to BB for your exercises.