Mathematics - Differentiation Question with Solution | TestHub

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

$\Rightarrow \frac{dy}{dx}=f^{\text{'}}\left(f\left(f\left(x\right)\right)\right)\times f^{\text{'}}\left(f\left(x\right)\right)\times f^{\text{'}}\left(x\right)+2f\left(x\right)\times f^{\text{'}}\left(x\right)$

$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=1}=f\text{'}\left(f\left(f\left(1\right)\right)\right)\times f^{\text{'}}\left(f\left(1\right)\right)\times f^{\text{'}}\left(1\right)+2f\left(1\right)\times f^{\text{'}}\left(1\right)$

$=f^{\text{'}}\left(f\left(1\right)\right)\times f^{\text{'}}\left(1\right)\times 3+2\times 1\times 3$

$=f^{\text{'}}\left(1\right)\times 3\times 3+6$

$=3\times 3\times 3+6$

$=27+6$

$=33$