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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.html7/30/2019 MFB Partial Differentation
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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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