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

 

Answers (1)

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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