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

Question

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

Accepted Answer

Let $F(x)=f(f(f(x)))+(f(x))^2$. $F'(x)=f'(f(f(x)))f'(f(x))f'(x)+2f(x)f'(x)$.

At $x=1$: $F'(1)=f'(f(f(1)))f'(f(1))f'(1)+2f(1)f'(1)=f'(1)f'(1)f'(1)+2(1)(3)=27+6=33$.

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