در این سؤال اتحاد ${{x}^{3}}-1=\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)$ به کار رفته است. $\begin{align}  & f\left( x \right)={{x}^{2}}\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)-{{x}^{5}}\Rightarrow f\left( x \right)={{x}^{2}}\left( {{x}^{3}}-1 \right)-{{x}^{5}} \\  & \Rightarrow f\left( x \right)=\cancel{{{x}^{5}}}-{{x}^{2}}-\cancel{{{x}^{5}}}=-{{x}^{2}}\Rightarrow {f}'\left( x \right)=-2x \\ \end{align}$