To find $F'(x)$, we can use the Fundamental Theorem of Calculus, which states that if

$F(x) = \int_{a}^{g(x)} f(t) \, dt$

then

$F'(x) = f(g(x)) \cdot g'(x)$

In this problem, $F(x) = \int_{0}^{\sin(x)} 5\sqrt{t} \, dt$. Here, $f(t) = 5\sqrt{t}$ and $g(x) = \sin(x)$.

First, find the derivative of $g(x) = \sin(x)$:

$g'(x) = \cos(x)$

Now, apply the Fundamental Theorem of Calculus:

$F'(x) = 5\sqrt{\sin(x)} \cdot \cos(x)$

So,

$F'(x) = 5\sqrt{\sin(x)} \cdot \cos(x)$