Mathematics: Lecture 12 ‫مدرس ازهار مالك‬

Partial Differentiations:

Partial Differentiations

Z  f ( x, y) or f(x,y,z)  0

Z 
 Zx  fx 
x 
 1st partial derivatives
Z 
 Zy  fy 
y 

2Z 
 Z xx  f xx 
x 2

2Z 
 Z yy  f yy 
y 2

 2nd partial derivatives
 Z
2

 Z yx
yx 

2Z 
 Z xy
xy 
Z xy  Z yx

Ex.1
Z Z
If Z  x y , find ,
x y
Z Z
 y x y -1 y constant ,  x y  ln x  dy , x constant  power function
x y

Ex.2
y
If Z  tan 1 , show that Zyx  Z xy
x
1 y y
Zx   2  2
y 2
x x  y2
1 2
x
( x 2  y 2 )( 1)  y.2 y y2  x2
Z yx   2  (1)
( x 2  y 2 )2 ( x  y 2 )2

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Mathematics: Lecture 12 ‫مدرس ازهار مالك‬
Partial Differentiations:

1 1 x
Zy    2
y x x  y2
2
1 2
x
( x  y 2 )(1)  x.2 x
2
y 2  x2
Z xy    (2)
( x 2  y 2 )2 ( x 2  y 2 )2

(1) & (2) are equal

Properties:

1) If   f ( ) ,   g( x, y)

   f  
  or  
x  x  x 

 chain rule
   
  
y  y 

2) If   f ( x, y) , x  g(r,s) , y  h(r,s) \

  x  y 
    
r x r y r


 chain rule
  x  y 
s    
r x s y s 


3) Total differential

If   f ( x, y, z,.....)
d  f x dx  f y dy  f z dz  ...
or d  x dx   y dy  z dz  ...

Ex.1
d
If   f ( x, y, z)  xyz  x2  y 2  z 2 , Find
dx
By property (3)

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Mathematics: Lecture 12 ‫مدرس ازهار مالك‬
Partial Differentiations:

d   x dx   y dy   z dz
d  ( yz  2 x)dx  ( xz  2 y )dy  ( xy  2 z )dz
d dx dy dz
 ( yz  2 x)  ( xz  2 y )  ( xy  2 z )
dx dx dx dx

Ex.2
 2 2  
2
If   f ( x  ct)  g ( x  ct) .....(1) , Show that c
t 2 x 2
There are two methods to solve this Ex.

First method:

Let x  ct  r , x  ct  s
Eq.(1) becomes

  f (r )  g ( s ) .....(1)

 f (r ) r g ( s ) s
   
t r t s t
 f (r )  c  g ( s )  (c)

 2  f (r ) r   g ( s) s 
 c    c  
t 2
 r t   s t 

 c f (r )  c   cg ( s)  (c) 

 2
  c 2  f (r )   g ( s)  ...... (2)
t 2

 f (r ) r g ( s) s
   
x r x s x
 f (r )  1  g ( s)  1

 2
 f (r )  1  1  g (r )  1  1
x 2

In eq.(2)

 2 2  
2
 c
t 2 x 2

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Mathematics: Lecture 12 ‫مدرس ازهار مالك‬
Partial Differentiations:

Second method:
t ‫) مباشرة بالنسبة لـ‬1( ‫نشتق معادلة‬


 f ( x  ct )  c  g ( x  ct)  (c)
t

 2
 f ( x  ct )  c  c  g ( x  ct )  (c)  (c)
t 2

 2
  c 2  f   g  ..... (2)
t 2


 f ( x  ct )  1  g ( x  ct )  1
x

 2
 f ( x  ct)  1  1  g ( x  ct)  1  1
x 2

 2
 2   f   g  ..... (3)
x

From (2) & (3)

 2 2  
2
 c
t 2 x 2

Ex.3
 y z z
If z  xn f   , Show that x  y  nz
x x y
z y y
x   x n 1  y f ( )  n x n  f ( ) ..... (1)
x x x

z y 1
 x n  f ( )  ( )  0
y x x
z y
y  x n 1  y f ( ) ..... (2)
y x

From (1) & (2)

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Mathematics: Lecture 12 ‫مدرس ازهار مالك‬
Partial Differentiations:

Ex.4
 
Express and in terms of r & s if   x  2 y  z 2 ,
r s
r
x , y  r 2  ln s , z  2r
s

  x  y  z
     
r x r y r z r
1 1 1
 1   2  2r  2 z  2   4r  4 z  12r 
s s s

  x  y  z
     
s x s y s z s
r 1 r 2
 1 2  2   2z  0  2 
s s s s

Problems:

f f f
Find , ,
x y z
y
1) f ( x, y, z )  z sin -1
x
x(2 - cos2y)
2) f ( x, y , z ) 
x 2  y2

3) Find when

y
u0 ,   0 if   x 2 , x  u  2  1 , y  2u   - 2
x
 xy   
4) If   f  2  , show that x
2 
y 0
x y  x y
5) If   f ( x, y) , and x  r cos , y  r sin  , show that
 1 
( ) 2  2 ( ) 2  f x2  f y2
x r y

dz
6) If f ( x, y, z )  0 & z  x  y , find
dx
y
7) Find the directional derivative of f ( x, y )  x tan 1 at (1,1) in the
x
  
direction of A  2i  j
x2  y 2
8) In which direction is the directional derivative of f ( x, y)  2
x  y2

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Mathematics: Lecture 12 ‫مدرس ازهار مالك‬
Partial Differentiations:

9) The D.D. of f ( x, y) at p0 (1,2) in the direction towards p1 (2,3) is

References:

1- calculus & Analytic Geometry (Thomas).