Answer
$F^{\prime}(x)=3x^{2}\sin x^{6}$
Work Step by Step
See Example 8.
Substituting $u=x^{3},\displaystyle \quad \frac{du}{dx}=3x^{2}$
Chain Rule:
$\displaystyle \frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}$
$=\displaystyle \frac{d}{du}[F(x)]\cdot\frac{du}{dx}$
$=\displaystyle \frac{d}{du}[\int_{0}^{x^{3}}\sin t^{2}dt]\cdot\frac{du}{dx}$
... apply the substitution ...
$=\displaystyle \frac{d}{du}[\int_{0}^{u}\sin t^{2}dt]\cdot 3x^{2}$
... apply the 2nd FTC,
$= \sin u^{2}\cdot 3x^{2}$
... bring x back ...
$= \sin(x^{3})^{2}\cdot 3x^{2}$
$F^{\prime}(x)=3x^{2}\sin x^{6}$
