Answers and Explanations 271
Answers
401–500
413.
−
− +
+
+
−
2
1
1
1
1
2
2
2
2
xy
x
y
xy
xy
x
x x
xy
xy
(
)
( ) ( )
(
) (
)
( ) ( )


sec
tan
sec
tan
 

Take the derivative of both sides of sec( )
xy
y
x
= +
1
2
and solve for
dy
dx
:
sec( )tan( ) ( )
sec
xy
xy
y x
dy
dx
x
dy
dx
y x
x
y
1
1
2
1
2
2
2
+
=
+
−
+

 

 
(
)
( )
(
)
( ( )tan( )
sec( )tan( )
xy
xy x
xy
xy
dy
dx
x
dy
dx
xy
x
dy
dx
x
+
(
)
= +
−
+
1
1
2
1
2
2
2
s sec( )tan( )
sec( )tan( )
xy
xy
x
xy
x
y
xy
xy
dy
d
− +
= −
+
−
1
1
2
1
2
2
2

 

 
(
)
x x
xy
x
y
xy
xy
x
xy
xy
x
dy
dx
=
−
+
−
− +
=
−
2
1
1
1
2
2
2
2
(
)
sec( )tan( )
sec( )tan( )
x xy
x
y
xy
xy
x
x
xy
xy
x
dy
d
− +
−
+
+
1
1
1
2
2
2
2
2
(
)
(
)
(
)
sec( )tan( )
sec( )tan( )
x x
xy
x
y
xy
xy
x
x x
xy
xy
=
−
− +
+
+
−
2
1
1
1
2
2
2
2
(
)
(
) (
)
sec( )tan( )
sec( )tan( ) 1 1




414.
−64
3
y
Begin by finding the first derivative of 8x
2
+ y
2
= 8:
16
2
0
2
1 6
8
x
y
dy
dx
y
dy
dx
x
dy
dx
x
y
+
=
= −
= −
Next, find the second derivative:
d y
dx
y
x
dy
dx
y
2
2
2
8
8
=
−
( ) (
)
− −