mfb partial differentation

7/30/2019 MFB Partial Differentation

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Partial Differentiation


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The ordinary derivative of a function ofone variable can be carried out

because everything else in the function is a constant and does not

affect the process of differentiation.

When there is more than one variable in a function it is often useful

to examine the variation of the function with respect to one of the

variables with all the other variables constrained to stay constant.

This is the purpose of a partial derivative.

For example, given the polynomial in variables x and y,

the partial derivative with respect to x is written

and the partial derivative with respect to y is written

The Partial Derivative
http://hyperphysics.phy-astr.gsu.edu/Hbase/deriv.htmlhttp://hyperphysics.phy-astr.gsu.edu/Hbase/deriv.html


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The ordinary derivative of a function of several variables with

respect to one of the independent variables, keeping all other

independent variables constant, is called the partial derivativeof the function.

Partial derivatives off(x,y) with respect to x and y are denoted by

x

f

y

f

By definitionx

yxfyxxf

x

f

x

),(),(lim

0

y

yxfyyxf

y

f

y

),(),(lim

0

When these limits exist


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Example:

2332),( xyxyxf

2236 yx

x

f

If

xyy

f6


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Higher order partial derivatives

Iff(x,y) has partial derivatives at each point (x.y) in a region, then and

are themselves fuctions of x and y, which may also have partial derivatives. These

second derivatives are denoted by

xf / yf /

2

2

x

f

x

f

x

2

2

y

f

y

f

y

2

2

z

f

z

f

z


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23

),(xy

eyxyx findIf

(a) (b) (c) (d)

(e) (f)

x

y

xx

yy

xy

2

yx

2

Example:

Solution:

(a) x

2

3 xy

eyxx

22

.3

2

yeyxxy

222

3xy

eyyx

y

23 xyeyxy

xyex xy 2.

23

2

.23 xy

exyx (b)


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02

2

2

2

2

2

z

U

y

U

x

U 2/1222,, zyxzyxUsuppose , show that

0,0,0,, zyxWe asume here that Then,

x

U

xzyx 2.2

1 2/3222 2/3222 zyxx

2

2

x

U

2/3222

zyxx

x

1.2.2

3 2/32222/5222

zyxxzyxx

2/5222222

2/5222

23

zyx

zyx

zyx

x

2/5222222

2

zyx

zyx


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2

2

y

U

2/5222222

2

zyx

zxy

2

2

z

U

2/5222222

2

zyx

yxz

2

2

2

2

2

2

z

U

y

U

x

U

Adding

2/5222222

2

zyx

zyx

2/5222222

2

zyx

zxy

2/5222222

2

zyx

yxz

2/5222

222222222222

zyx

yxzzxyzyx

2/5222222222

222222

zyx

zzyyxx

0


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