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L E C T U R E 1 0
H i g h e r O r d e r D e r i v a t i v e s a n d T a y l o r E
x p a n s i o n s
1 . H i g h e r O r d e r D e r i v a t i v e s
S i n c e a p a r t i a l d e r i v a t i v e o f a f u n c t i
o n f : R
n
R i s ( w h e r e v e r i t e x i s t s ) a g a i n a f u n c t
i o n f r o m R
n
t o R i t
m a k e s s e n s e t o t a l k a b o u t p a r t i a l d e r i
v a t i v e s o f p a r t i a l d e r i v a t i v e s ; i . e . , h
i g h e r o r d e r p a r t i a l d e r i v a t i v e s .
E x a m p l e 1 0 . 1 . C o m p u t e
2
f
x
2
x
f
x
,
2
f
x y
x
f
y
a n d
2
f
y x
y
f
x
w h e r e f ( x , y ) = 3 x
2
y + x
2
.
2
f
x
2
x
f
x
=
x
( 6 x y + 2 x )
= 6 y + 2
2
f
x y
x
f
y
=
x
3 x
2
+ 0
= 6 x
2
f
y x
y
f
x
=
y
( 6 x y + 2 x )
= 6 x + 0
= 6 x
N o t e t h a t i n t h i s e x a m p l e
2
f
x y
=
2
f
y x
T h i s i s i n f a c t a g e n e r a l p h e n o m e n o n ; t
h e v a l u e o f a m i x e d p a r t i a l d e r i v a t i v e d o
e s n o t d e p e n d o n t h e o r d e r
i n w h i c h t h e d e r i v a t i v e s a r e t a k e n . S t
a t e d m o r e f o r m a l l y ;
T h e o r e m 1 0 . 2 . I f f : R
n
R i s s u c h t h a t a l l d o u b l e p a r t i a l d e r i v
a t i v e s
2
f
x
i
x
j
e x i s t a n d a r e c o n t i n o u s ,
t h e n
2
f
x
i
x
j
=
2
f
x
j
x
i
1
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2 . T A Y L O R S F O R M U L A F O R F U N C T I O N S O F S E
V E R A L V A R I A B L E S 3
W e h a v e
f ( 1 , 1 ) = 1 + 1 + 1 = 3
f
y
( 1 , 1 )
= ( y + 2 x + 0 ) |
( 1 , 1 )
= 3
f
y
( 1 , 1 )
= ( x + 0 + 2 y ) |
( 1 , 1 )
= 3
2
f
x
2
( 1 , 1 )
= ( 0 + 2 + 0 ) |
( 1 , 1 )
= 2
2
f
x y
( 1 , 1 )
=
2
f
y x
( 1 , 1 )
= ( 1 + 0 + 0 ) |
( 1 , 1 )
= 1
2
f
y
2
( 1 , 1 )
= ( 0 + 0 + 2 ) |
( 1 , 1 )
= 2
S o
f ( x , y ) = f ( 1 , 1 ) +
f
y
( 1 , 1 )
( x 1 ) +
f
y
( 1 , 1 )
( y 1 )
+
1
2
2
f
x
2
( 1 , 1 )
( x 1 )
2
+
2
f
x y
( 1 , 1 )
( x 1 ) ( y 1 )
+
2
f
y x
( 1 , 1 )
( y 1 ) ( x 1 ) +
2
f
y
2
( 1 , 1 )
( y 1 )
2
+ O
( x , y ) ( 1 , 1 )
3
= 3 + 3 ( x 1 ) + 3 ( y 1 ) +
1
2
2 ( x 1 )
2
+ 2 ( x 1 ) ( y 1 ) + 2 ( y 1 )
2
+ O
( x , y ) ( 1 , 1 )
3
= 3 + 3 ( x 1 ) + 3 ( y 1 ) + ( x 1 )
2
+ ( x 1 ) ( y 1 ) + ( y 1 )
2
+ O
( x , y ) ( 1 , 1 )
3
B e l o w I p r e s e n t a n o t h e r ( e q u i v a l e n t )
f o r m u l a f o r t h e s e c o n d o r d e r T a y l o r e x p a
n s i o n .
L e t ( x a ) b e t h e n - d i m e n s i o n a l c o l u m n v
e c t o r w i t h c o m p o n e n t s
( x a ) =
x
1
a
1
x
2
a
2 1
.
.
.
x
n
a
n
a n d l e t ( x a )
T
b e t h e m a t r i x t r a n s p o s e o f ( x a ) ( a n n - d
i m e n s i o n a l r o w v e c t o r )
( x a )
T
= ( x
1
a
1
, x
2
a
2
, , x
n
a
n
) .
T h e g r a d i e n t v e c t o r f ( a ) = D f ( a ) , a c c o
r d i n g t o t h e c o n v e n t i o n s o f S e c t i o n 2 . 3 i
s a n n - d i m e n s i o n a l r o w
v e c t o r ;
f ( a ) =
f
x
1
( a ) ,
f
x 2
( a ) , ,
f
x
n
( a )
.
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2 . T A Y L O R S F O R M U L A F O R F U N C T I O N S O F S E
V E R A L V A R I A B L E S 4
L e t u s n o w d e fi n e t h e H e s s i a n m a t r i x a t t
h e p o i n t a a s t h e n n m a t r i x H f ( a ) d e fi n e d b
y
H f ( a ) =
2
f
x
1
x
1
( a )
2
f
x
1
x
2
( a )
2
f
x
1
x
n
( a )
2
f
x
2
x
1
( a )
2
f
x
2
x
2
( a )
.
.
.
.
.
.
.
.
.
.
.
.
2
f
x
n
x
1
( a )
2
f
x
n
x
n
( a )
.
T h e n w e c a n w r i t e
f ( x ) f ( a ) + f ( a ) ( x a ) +
1
2
( x a )
T
H f ( a ) ( x a ) + O
x a
3
f o r t h e s e c o n d o r d e r T a y l o r e x p a n s i o n
o f f a b o u t a .
