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JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Differentiation

Let

f

be a differentiable function such that

f (1) = 2

and

f^{'} (x) = f (x)

for all

x \in R

. If

h (x) = f (f (x))

, then

h^{'} (1)

is equal to :

(A) $2 e^{2}$
(B)4 e✓ Correct
(C)2 e
(D) $4 e^{2}$

Answer: (B)

Solution

From

f^{'} (x) = f (x)

,

f (x) = A e^{x}

, and

f (1) = 2

fixes

A = 2 e^{- 1}

, so

f (x) = 2 e^{x - 1}

and

f^{'} (x) = 2 e^{x - 1}

. For

h (x) = f (f (x))

the chain rule gives

h^{'} (x) = f^{'} (f (x)) f^{'} (x)

, so

h^{'} (1) = f^{'} (f (1)) f^{'} (1) = f^{'} (2) f^{'} (1) = (2 e) (2) = 4 e

.

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Single correct option

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