Matakuliah
Tahun
: K0124 / Matematika Teknik II
: 2006/2007
PERTEMUAN 1
PARTIAL DERIVATIVES OF
A FUNCTION OF TWO
VARIABLES
1
Partial derivatives
Given z  f ( x, y ) as a function of two independent variables x and y.
If x varies but y is fixed, then z is a function of x and the derivative
of z with respect to x is given by
f x ( x, y ) 
z
f ( x  x, y )  f ( x, y )
 lim
x x0
x
and it is called the partial derivative of z  f ( x, y ) with respect to x.
2
Conversely, if y varies but x is fixed, then z is a function of y
and the derivative of z with respect to y is given by
z
f ( x, y  y)  f ( x, y)
f y ( x, y)   lim
y y0
y
and it is called the partial derivative of z  f ( x, y ) with respect to y.
3
Example 13.1
2
2
z

f
(
x
,
y
)

2
x

3
xy

4
y
. Then,
Given
z
z
fx 
 4 x  3 y, f y 
 3x  8 y.
x
y
4
Example 13.2
Given
x2 y2
z  g ( x, y ) 

.
y
x
Then,
z
x y2
z
x2
y
gx 
 2  2 , gy 
 2 2 .
x
y x
y
y
x
5
Example 13.3
Given
z  h( x, y)  sin(2 x  3 y).
Then,
z
z
hx 
 2 cos( 2 x  3 y ), hy 
 3 cos( 2 x  3 y ).
x
y
6
Partial derivatives
of higher orders
z
The partial derivative
of
x
z  f ( x, y)
can be differentiated partially
with respect to x and y, yielding the second partial derivatives
2 z
  z   2 z
  z 
 f xx ( x, y )   ,
 f xy ( x, y )   .
2
x
x  x  yx
y  x 
7
z
Also, from
can be obtained
y
2 z
  z   2 z
  z 
 f yx ( x, y)   , 2  f yy ( x, y)   .
xy
x  y  y
y  y 
If z  f ( x, y ) and its partial derivatives are continuous, it
can be proved that the order of differentiation is immaterial, i.e.
2 z
2 z

.
xy yx
8
Example 13.4
2
2
z

f
(
x
,
y
)

x

3
xy

y
 5. Then,
Given
z
z
2z
2z
2z
 2 x  3 y,  3x  2 y,

 3, 2  2.
x
y
xy yx
y
9
Example 13.5
Given
z  g ( x, y )  e
x 2  xy
.
Then,
z
x 2  xy z
x 2  xy
 ( 2 x  y )e
,  xe
,
x
y
2
2
2
2z

z
x  xy
2 x  xy
2 x 2  xy
 2e
 (2 x  y ) e
, 2 x e
,
2
x
y
2
2 z
x 2  xy
x 2  xy  z
x 2  xy
x 2  xy
e
 x ( 2 x  y )e
,
e
 x ( 2 x  y )e
.
yx
xy
10
Partial derivatives
of implicit functions
If z is defined implicitly as a function of independent variables x and y by the equation
F ( x, y, z )  0, then the partial derivatives
z
z
and
can be determined by the method
x
y
shown in the following examples.
11
Example 13.6
Find
z
z
and
of the implicit function x 2  y 2  z 2  25.
x
y
 ( x 2  y 2  z 2 )  (25)
z
z
x

,2 x  2 z  0,   .
x
x
x
x
z
Similarly,
 ( x 2  y 2  z 2 )  (25)
z
z
y

,2 y  2 z  0,   .
y
y
y
y
z
12
TERIMA KASIH
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