Section 13.3 Partial Derivatives
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Transcript of Section 13.3 Partial Derivatives
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Def’n
0
0
, ,, lim
, ,, lim
xh
yh
f x h y f x yf x y
hf x y h f x y
f x yh
In the first, we are holding y constant, finding only how the level curve is changing as x is changing.
In the second, we are holding x constant, finding how the level curve changes as y is changing.
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Notation
, ,
, ,
x x
y y
f zf x y f f x y
x x xf z
f x y f f x yy y y
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Rules for Finding Partial Derivatives
To find , regard as a constant and
differentiate , with respect to .
To find , regard as a constant and
differentiate , with respect to .
x
y
f y
f x y x
f x
f x y y
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Example
2 2If , 4 2 ,
find 1,1 and 1,1 and
interpret these numbers as slopes.
x y
f x y x y
f f
, 2
1,1 2 1 2
, 4
1,1 4 1 4
x
x
y
y
f x y x
f
f x y y
f
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Example
Find , , and if , , ln
ln
ln
1
xyx y z
xyx
xyy
xyxy
z
f f f f x y z e z
f z e y
f z e x
ef e
z z
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Higher Derivatives
2 2
2 2
2 2
2 2
2 2
2 2
x xxx
x xyy
y yxx
y yyy
f f zf f
x x x x
f f zf f
y x y x y x
f f zf f
x y x y x y
f f zf f
y y y y
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Example 3 2 3 2
2 3
2 2
3
2
2
2
, 2
3 2
3 4
6 2
6 4
6
6
x
y
xx
yy
xy
yx
f x y x x y y
f x xy
f x y y
f x y
f x y
f xy
f xy
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Clairaut’s Theorem
, ,xy yx
xyz xzy zyx zxy yzx yxz
f x y f x y
f f f f f f
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Assignment
13.3, 5-39 odd, 45-65 odd