Aus Online Mathematik Brückenkurs 2

With the help of the square rule, we can expand the quadratic as

\displaystyle \begin{align} & f\left( x \right)=\left( x^{2}-1 \right)^{2}=\left( x^{2} \right)^{2}-2\centerdot x^{2}\centerdot 1+1^{2} \\ & =x^{4}-2x^{2}+1 \\ \end{align}

When the function is written in this form, it is easy to differentiate term by term:

\displaystyle \begin{align} & {f}'\left( x \right)=\frac{d}{dx}\left( x^{4}-2x^{2}+1 \right) \\ & =\frac{d}{dx}x^{4}-2\frac{d}{dx}x^{2}+\frac{d}{dx}1 \\ & =4\centerdot x^{-1}-2\centerdot 2x^{-1}+0 \\ & =4x^{3}-4x=4x\left( x^{2}-1 \right) \\ \end{align}