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Byju's Answer
Standard XII
Mathematics
Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions
Differentiate...
Question
Differentiate the following functions w.r.t.x
tan
−
1
(
√
1
+
a
2
x
2
a
x
)
Open in App
Solution
Let
y
=
t
a
n
−
1
(
√
1
+
a
2
x
2
a
x
)
t
a
n
y
=
√
1
+
a
2
x
2
a
x
⇒
t
a
n
2
y
=
1
+
a
2
x
2
a
2
x
2
a
2
x
2
t
a
n
2
y
=
1
+
a
2
x
2
by squaring both sides
a
2
x
2
(
t
a
n
2
y
−
1
)
=
1
a
2
[
x
2
2
t
a
n
y
s
e
c
2
y
d
y
d
x
+
2
x
(
t
a
n
2
y
−
1
)
]
=
0
⇒
2
a
2
x
2
t
a
n
y
s
e
c
2
y
.
d
y
d
x
=
2
x
(
1
−
t
a
n
2
y
)
⇒
a
2
x
√
1
+
a
2
x
2
a
x
s
e
c
2
y
d
y
d
x
=
1
−
1
+
a
2
x
2
a
2
x
2
⇒
a
√
1
+
a
2
x
2
(
1
+
t
a
n
2
y
)
d
y
d
x
=
a
2
x
2
−
1
−
a
2
x
2
a
2
x
2
⇒
(
1
+
1
+
a
2
x
2
a
2
x
2
)
d
y
d
x
=
−
1
a
2
x
2
×
a
√
1
+
a
2
x
2
⇒
2
a
2
x
2
+
1
a
2
x
2
d
y
d
x
=
−
1
a
2
x
2
×
a
√
1
+
a
2
x
2
⇒
d
y
d
x
=
−
1
a
2
x
2
×
a
√
1
+
a
2
x
2
⇒
d
y
d
x
=
−
1
a
2
x
2
×
a
√
1
+
a
2
x
2
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