(b) The first derivative of
1
2
2
( )
7
3
2
g x
x
x
=
+
−
is
1
2
( )
6
dg x
x
x
dx
−
=
−
. Thus, the
first derivative of
( )
g x
when
x
=
4 would be
4
2
(
7
)
4
x
dg x
dx
=
=
.
4.
Product rule:
( )
( )
( )
g x
h x
k x
=
⋅
( )
'( )
( )
( )
( )
g
x
h
x
k x
h x
k
x
′
′
⇒
=
⋅
+
⋅
.
for this question
3
( )
ln( )
g x
x
x
=
⋅
,where
3
( )
h x
x
=
and
( )
ln( )
k x
x
=
thus,
2
3
2
2
( )
( )
( )
( )
( )
3
ln( )
(1/
)
3
ln( )
g
x
h x
k x
h x
k
x
x
x
x
x
x
x
x
′
′
′
=
⋅
+
⋅
=
⋅
+
⋅
=
+
5.
Quotient rule:
2
( )
(
)
( )
'( )
'( )
( )
( )
:
( )
f
y
d
g y f
y
g
y f
y
g y
dy
g
y
−
=
For this question:
( )
3
2
'( )
3
( )
5
7
'( )
5
f
y
y
f
y
g y
y
g
y
=
+
=
⎧
⎧
⇒
⎨
⎨
=
−
=
⎩
⎩
Hence,
2
2
2
( )
'( )
'( )
( )
(5
7)
3
(3
2)
5
31
( )
(5
7)
25
70
49
g y f
y
g
y f
y
y
y
g
y
y
y
y
−
−
×
−
+
×
−
=
=
−
−
+

3
6.
Chain rule:
( )
:
df z
df
dy
dz
dy
dz
=
×
,
For this question:
2
2
3
1
2
3
2
1
(2
12
)
3
( )
(2
12
)
(2
12
)
(2
24 )
df
z
z
dy
f z
z
z
dy
y
z
z
z
dz
−
⎧
=
+
⎧
⎪
⎪
⎪
=
+
⇒
⎨
⎨
⎪
⎪
=
+
⎩
=
+
⎪
⎩
Hence:
2
2
2
3
2
3
( )
1
(2
24 )(2
12
)
:
(2
12
)
(2
24 )
3
3
df z
df
dy
z
z
z
z
z
z
dz
dy
dz
−
−
+
+
=
×
=
+
×
+
=
Below is the graph of
2
10
y
x
+
=
,we plot it as
1
5
2
y
x
=
−
. The area under the line
in the positive quadrant is the area of a right triangle with a base of 10 and a height of
5
10
5 / 2
25
⇒
×
=
Alternatively, we can use calculus, the area equals to the integration of y with

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- Spring '12
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- Microeconomics, Calculus, Derivative, red line, dy dz, 2 2 2 g, df dy