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JEE Main Math Class 12 Continuity and Differentiability Part 1 Questions
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© examsnet.com
Question : 16
Total: 100
Let
f
be a differentiable function such that
f
(
1
)
=
2
and
f
′
(
x
)
=
f
(
x
)
for all
x
∈
R
. If
h
(
x
)
=
f
(
f
(
x
)
)
,
then
h
′
(
1
)
is equal to :
[Jan. 12, 2019 (II)]
2
e
2
4
e
2
e
4
e
2
Validate
Solution:
Since,
f
′
(
x
)
=
f
(
x
)
Then,
f
′
(
x
)
f
(
x
)
=
1
⇒
f
′
(
x
)
f
(
x
)
=
d
x
⇒
f
′
(
x
)
f
(
x
)
d
x
=
∫
d
x
⇒
ln
|
f
(
x
)
|
=
x
+
c
f
(
x
)
=
±
e
x
+
c
.....(1)
Since, the given condition
f
(
1
)
=
2
From eq
n
(
1
)
f
(
x
)
=
e
x
+
c
=
e
c
e
x
Then,
f
(
1
)
=
e
c
.
e
1
⇒
2
=
e
c
.
e
⇒
2
e
=
e
c
Then, from eq
n
(
1
)
f
(
x
)
=
2
e
e
x
⇒
f
′
(
x
)
=
2
e
e
x
Now
h
(
x
)
=
f
(
f
(
x
)
)
⇒
h
′
(
x
)
=
f
′
(
f
(
x
)
)
.
f
′
(
x
)
h
′
(
1
)
=
f
′
(
2
)
.
f
′
(
1
)
=
2
e
e
2
.
2
e
.
e
=
4
e
© examsnet.com
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