Section 13.3Partial Derivatives

Calculus IIINovember 3, 2009Berkley High Schooltodd1@toddfadoir.com

2

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.

3

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

4

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

5

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

6

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

7

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

8

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

9

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