Answer
$4x^{3}-2x$
Work Step by Step
One way: use the product rule.
Another way: simplify the function
(as a difference of squares) ,
$y=x^{4}-x^{2}$ and use the power and sum rules:
$\displaystyle \frac{dy}{dx}=4x^{3}-2x$
With the product rule,
$f(x)=x^{2}+x,\quad f^{\prime}(x)=2x+1$
$g(x)=x^{2}-x,\quad g^{\prime}(x)=2x-1$
$\displaystyle \frac{d}{dx}[f(x)g(x)]=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$.
$\displaystyle \frac{dy}{dx}=(2x+1)(x^{2}-x)+(x^{2}+x)(2x-1)$
$=2x^{3}-2x^{2}+x^{2}-x+2x^{3}-x^{2}+2x^{2}-x$
$=4x^{3}-2x$
