Answer
$F^{\prime}(x)=2x\sin x^{4}$
Work Step by Step
See Example 8.
Substituting $u=x^{2},\displaystyle \quad \frac{du}{dx}=2x$
$\displaystyle \frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}$
$=\displaystyle \frac{d}{du}[F(x)]\cdot\frac{du}{dx}$
$=\displaystyle \frac{d}{du}[\int_{0}^{x^{2}}\sin\theta^{2}d\theta]\cdot\frac{du}{dx}$
... apply the substitution ...
$=\displaystyle \frac{d}{du}[\int_{0}^{u}\sin\theta^{2}d\theta]\cdot 2x$
... apply the 2nd FTC, $\displaystyle \frac{d}{dx}[\int_{a}^{x}f(t)dt]=f(x)$.
$= \sin u^{2}\cdot 2x$
... bring x back ...
$= \sin(x^{2})^{2}\cdot 2x$
$F^{\prime}(x)=2x\sin x^{4}$
