If f(1)=1,

Question:

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^{2}$ at $x=1$ is :

  1. (1) 33

  2. (2) 12

  3. (3) 15

  4. (4) 9


Correct Option: 1

Solution:

Let $g(x)=f(f(f(x)))+(\mathrm{f}(x))^{2}$

Differentiating both sides w.r.t. $x$, we get

$g^{\prime}(x)=f^{\prime}(f(f(x))) f^{\prime}(f(x)) f^{\prime}(x)+2 f(x) f^{\prime}(x)$

$g^{\prime}(1)=f^{\prime}(f(f(1))) f^{\prime}(f(1)) f^{\prime}(1)+2 f(1) f^{\prime}(1)$

$=f^{\prime}(f(1)) \mathrm{f}^{\prime}(1) f^{\prime}(1)+2 f(1) f^{\prime}(1)$

$=3 \times 3 \times 3+2 \times 1 \times 3=27+6=33$

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