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Let f be a differentiable function such ...

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

`2e^(2)`

B

2e

C

4e

D

`4e^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`f'(x) = f(x)` Integrating both side
`int(dy)/(dX) = int y rArr int(dy)/(y) = int dx rArr ln y = x + c rArr y = e^(x).c`
As f(1) = 2, `2 = e.c rArr c = (2)/(e)`
`y = e^(x)(2)/(e) = 2.e^(x-1) = f(x)`
`h(x) = f(f(x)), h'(x) = f'(f(x))f'(x)`
`rArr h'(1) = f'(f(1)).f(1) = f'(2) xx f'(1) = 2e xx 2 = 4e`
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