Answer
$g'(x)=6x(x^{2}+1)^{2}(x^{2}+2)^{5}(3x^{2}+4)$
Work Step by Step
$g(x)=(x^{2}+1)^{3}(x^{2}+2)^{6}$
Differentiate using the product rule:
$g'(x)=[(x^{2}+1)^{3}][(x^{2}+2)^{6}]'+[(x^{2}+2)^{6}][(x^{2}+1)^{3}]'=...$
Use the chain rule to find $[(x^{2}+2)^{6}]'$ and $[(x^{2}+1)^{3}]'$:
$...=[(x^{2}+1)^{3}][6(x^{2}+2)^{5}(x^{2}+2)']+[(x^{2}+2)^{6}][3(x^{2}+1)^{2}(x^{2}+1)']=...$
$...=[(x^{2}+1)^{3}][6(x^{2}+2)^{5}(2x)]+[(x^{2}+2)^{6}][3(x^{2}+1)^{2}(2x)]=...$
$...=12x(x^{2}+1)^{3}(x^{2}+2)^{5}+6x(x^{2}+2)^{6}(x^{2}+1)^{2}$
Take out common factors $6x$, $(x^{2}+1)^{2}$ and $(x^{2}+2)^{5}$ to present a better looking answer (Optional)
$g'(x)=6x(x^{2}+1)^{2}(x^{2}+2)^{5}[2(x^{2}+1)+(x^{2}+2)]=...$
$...=6x(x^{2}+1)^{2}(x^{2}+2)^{5}(3x^{2}+4)$
