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4.16
The directional derivative
⃗
?
x
±
(
x
0
) measures the rate of increase of
±
in the di
rection
x
. Using Exercises 4.10, 4.14 and 3.61, assuming
x
has unit norm,
⃗
?
x
±
(
x
0
)=
?±
[
x
0
](
x
<
∇
±
(
x
0
)
,
x
>
≤
∪
∪
∇
±
(
x
0
)
∪
∪
This bound is attained when
x
=
∇
±
(
x
0
)
/
∪
∪
∇
±
(
x
0
)
∪
∪
since
⃗
?
x
±
(
x
0
<
∇
±
(
x
0
)
,
∇
±
(
x
0
)
∥∇
±
(
x
0
)
∥
>
=
∪
∪
∇
±
(
x
0
)
∪
∪
2
∥∇
±
(
x
0
)
∥
=
∪
∪
∇
±
(
x
0
)
∪
∪
The directional derivative is maximized when
∇
±
(
x
0
)and
x
are aligned.
4.17
Using Exercise 4.14
²
=
{
x
∈
³
:
<
∇
±
[
x
0
]
,
x
>
=0
}
4.18
Assume each
±
?
is diFerentiable at
x
0
and let
[
x
0
]=(
1
[
x
0
]
,?±
2
[
x
0
]
,...,?±
±
[
x
0
])
Then
f
(
x
0
+
x
)
−
f
[
x
0
]
−
?
f
[
x
0
]
x
=
⎛
⎜
⎜
⎜
⎝
±
1
(
x
0
+
x
)
−
±
1
[
x
0
]
−
1
[
x
0
]
x
±
2
(
x
0
+
x
)
−
±
2
[
x
0
]
−
2
[
x
0
]
x
.
.
.
±
±
(
x
0
+
x
)
−
±
±
(
x
0
)
−
±
[
x
0
]
x
⎞
⎟
⎟
⎟
⎠
and
±
?
(
x
0
+
x
)
−
±
?
(
x
0
)
−
?
[
x
0
]
x
∥
x
∥
→
0as
∥
x
∥→
0
for every
´
implies
f
(
x
0
+
x
)
−
f
(
x
0
)
−
?
f
[
x
0
](
x
)
∥
x
∥
→
∥
x
0
(4.43)
Therefore
f
is diFerentiable with derivative
?
f
[
x
0
]=
µ
=(
1
(
x
0
)
2
[
x
0
]
±
[
x
0
])
Each
?
[
x
0
] is represented by the gradient
∇
±
?
[
x
0
] (Exercise 4.13) and therefore
[
x
0
] is represented by the matrix
¶
=
⎛
⎜
⎜
⎜
⎝
∇
±
1
[
x
0
]
∇
±
2
[
x
0
]
.
.
.
∇
±
±
[
x
0
]
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
?
²
1
±
1
[
x
0
]
?
²
2
±
1
[
x
0
]
...
?
²
?
±
1
[
x
0
]
?
²
1
±
2
[
x
0
]
?
²
2
±
2
[
x
0
]
?
²
?
±
2
[
x
0
]
.
.
.
.
.
.
.
.
.
.
.
.
?
²
1
±
±
[
x
0
]
?
²
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.
 Fall '10
 Dr.DuMond
 Macroeconomics

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