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Mathematics
Problem
Let
g
(
x
)
=
arcsin
(
arctan
(
3
x
)
)
g(x)=\arcsin(\arctan(3x))
g
(
x
)
=
arcsin
(
arctan
(
3
x
))
. Which of the following gives
g
′
(
x
)
g'(x)
g
′
(
x
)
for all
x
x
x
in the domain of
g
g
g
? a)
3
1
+
9
x
2
1
−
(
arctan
(
3
x
)
)
2
\displaystyle \frac{3}{1+9x^{2}}\sqrt{1-(\arctan(3x))^{2}}
1
+
9
x
2
3
1
−
(
arctan
(
3
x
)
)
2
b)
3
1
−
(
3
x
)
2
(
1
+
9
x
2
)
\displaystyle \frac{3}{\sqrt{1-(3x)^{2}}\,(1+9x^{2})}
1
−
(
3
x
)
2
(
1
+
9
x
2
)
3
c)
3
(
1
+
9
x
2
)
1
−
(
arctan
(
3
x
)
)
2
\displaystyle \frac{3}{(1+9x^{2})\sqrt{1-(\arctan(3x))^{2}}}
(
1
+
9
x
2
)
1
−
(
arctan
(
3
x
)
)
2
3
d)
3
1
+
9
x
2
[
1
−
(
arctan
(
3
x
)
)
2
]
\displaystyle \frac{3}{\sqrt{1+9x^{2}}\,[1-(\arctan(3x))^{2}]}
1
+
9
x
2
[
1
−
(
arctan
(
3
x
)
)
2
]
3
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