#### Answer

f'(x)=10$x^{4}$+3$x^{2}$

#### Work Step by Step

The product rule states that if f(x)=h(x)g(x), then f'(x)=g'(x)h(x)+g(x)h'(x). We can find the derivative of the function f(x)=$x^{3}$($2x^{2}+1$) by setting g(x)=$x^{3}$ and h(x)=$2x^{2}+1$, and applying the product rule.
f'(x)=$\frac{d}{dx}$[$x^{3}$]($2x^{2}+1$)+($x^{3}$)$\frac{d}{dx}$[$2x^{2}+1$]
$\frac{d}{dx}$[$x^{3}$]=3$x^{2}$, using the power rule
$\frac{d}{dx}$[$2x^{2}+1$]=4x, using the power rule
Therefore,
f'(x)=(3$x^{2}$)($2x^{2}+1$)+($x^{3}$)(4x)
=(6$x^{4}$+3$x^{2}$)+4$x^{4}$ (Simplify)
=10$x^{4}$+3$x^{2}$ (Simplify)