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Joint Entrance Examination
Mathematics
Limits, Continuity and Differentiability
Question 195
Question 195 of 401
Medium
MCQ
Let
f
f
f
be a differentiable function such that
f
(
1
)
=
2
f(1)=2
f
(
1
)
=
2
and
f
′
(
x
)
=
f
(
x
)
f'(x)=f(x)
f
′
(
x
)
=
f
(
x
)
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
. If
h
(
x
)
=
f
(
f
(
x
)
)
h(x)=f(f(x))
h
(
x
)
=
f
(
f
(
x
))
, then
h
′
(
1
)
h'(1)
h
′
(
1
)
is equal to:
A
2
e
2
2e^2
2
e
2
B
4
e
4e
4
e
C
2
e
2e
2
e
D
4
e
2
4e^2
4
e
2
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Official solution
\[{{f'(x)} \over {f(x)}} = 1\forall x \in R\] Intergrate \& use f(1) = 2 f(x) = 2ex-1 \[ \Rightarrow \] f '(x) = 2ex-1 h(x) = f(f(x)) \[ \Rightarrow \] h'(x) = f '(f(x)) f'(x) h'(1) = f '(f(1)) f'(1) = f '(2) f '(1) = 2e . 2 = 4e
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