If $f(1)=1,f{}'(1)=3,$ then the derivative of $f\left ( f\left ( f\left ( x \right ) \right ) \right )+\left ( f(x) \right )^{2}$ at $x=1$ is: Option 1) $33$ Option 2) $12$ Option 3) $15$ Option 4) $9$

$y=f\left ( f\left ( f\left ( x \right ) \right ) \right )+\left ( f\left ( x \right ) \right )^{2}$

$\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}={f}'\left ( f\left ( f\left ( x \right ) \right ) \right )\times {f}'\left ( f\left ( x \right ) \right )\times {f}'\left ( x \right )+2f\left ( x \right )\times {f}'\left ( x \right )$

i$\frac{\mathrm{d} y}{\mathrm{d} x}|_{x=1}={f}'\left ( f\left ( f\left ( 1 \right ) \right ) \right )\times {f}'\left ( f\left ( 1 \right ) \right )\times {f}'\left ( 1 \right )+2f(1)\times {f}'\left ( 1 \right )$

given $f(1)=1,{f}'(1)=3$

$\frac{\mathrm{d} y}{\mathrm{d} x}|_{x=1}={f}'\left ( f\left ( 1 \right ) \right )\times {f}'\left ( 1 \right )\times 3+2\times 1\times 3$

$\frac{\mathrm{d} y}{\mathrm{d} x}|_{x=1}={f}'\left ( 1 \right )\times 3\times 3+6=33$

Option 1)

$33$

Option 2)

$12$

Option 3)

$15$

Option 4)

$9$

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