Answer:
Differentiate both sides of the equation.
d
d
x
(
tan
-1
(
5
x
2
y
)
)
=
d
d
x
(
x
+
4
x
y
2
)
Differentiate the left side of the equation.
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(
x
2
d
d
x
[
y
]
+
2
y
x
)
5
1
+
25
x
4
y
2
Differentiate the right side of the equation.
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4
y
2
+
8
x
y
d
d
x
[
y
]
+
1
Reform the equation by setting the left side equal to the right side.
(
x
2
y
'
+
2
y
x
)
(
5
1
+
25
x
4
y
2
)
=
4
y
2
+
8
x
y
y
'
+
1
Solve for
y
'
.
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y
'
=
10
y
x
−
4
y
2
−
100
y
4
x
4
−
1
−
25
x
4
y
2
x
(
8
y
+
200
x
4
y
3
−
5
x
)
Replace
y
'
with
d
y
d
x
.
d
y
d
x
=
10
y
x
−
4
y
2
−
100
y
4
x
4
−
1
−
25
x
4
y
2
x
(
8
y
+
200
x
4
y
3
−
5
x
)
