Domain of a any cubic function [tex]f(x)=ax^3+bx^2+cx+d[/tex] is defined to be always [tex]\mathbb{R}[/tex].
The derivative with respect to x of your cubic function is,
[tex]\dfrac{d}{dx}f(x)=f'(x)[/tex]
to find the derivative of a polynomial function, simply take a derivative of each factor and sum them up,
[tex]\dfrac{d}{dx}x^3=3x^2[/tex] by the rule [tex]\dfrac{d}{dx}x^m=mx^{m-1}[/tex] where [tex]m\in\mathbb{R}[/tex]
[tex]\dfrac{d}{dx}-6x=-6[/tex]
[tex]\dfrac{d}{dx}3=0[/tex]
So the derivative is,
[tex]f'(x)=3x^2-6[/tex]
both derivative and the original function have equal domain.
Hope this helps :)
