Suppose that you are told that the Taylor series of $f(x)=x^{2}e^{x^{3}}$ about $x=0$ is $$x^{2}+x^{5}+\frac{x^{8}}{2!}+\frac{x^{11}}{3!}+\frac{x^{14}}{4!} +\cdots.$$ Find each of the following:
$\frac{d}{dx}\left( x^{2}e^{x^{3}}\right)\bigg|_{x=0} =$
$\frac{d^{8}}{dx^{8}}\left( x^{2}e^{x^{3}}\right)\bigg|_{x=0} =$