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If f(1) = 1, f'(1) = 3, then the de...

If f(1) = 1, f'(1) = 3, then the derivative of f(f(x))) + `(f(x))^(2) ` at x = 1 is

A

12

B

9

C

15

D

33

Text Solution

Verified by Experts

The correct Answer is:
D
Let ` y = f(f(f(x))) + (f(x))^(2)`
On differentiating both sides w.r.t.x., we get
`(dy)/(dx) = f'(f(f(x)))*f'(f(x))* f'(x) + 2 f(x)f'(x)`
[by chain rule]
So, `(dy)/(dx)underset("at "x=1)(|) =f'(f(f(1)))*f'(f(1))*f'(1)+2f(1)f'(1)`
`:. (dy)/(dx)underset(x =1)(|)=f'(f(1))*f'(1)*(3)+2(1)(3)`
`[:' f(1)=1 and f'(1)=3]`
` =f'(1)*(3)*(3)+6`
`=(3xx9)+6=27+6 = 33`
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