**Solution.**To determine the derivative F(x) = f(g(x)) = \sin(\ln(x)), we will use the chain rule:

\begin{align*} F'(x) = f'(g(x))g'(x), \end{align*}

\begin{align*} f'(g(x)) = \cos(g(x)) = \cos(\ln(x)). \end{align*}

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= \cos(\ln(x))\frac{1}{x} \\ &= \frac{\cos(\ln(x))}{x}. \end{align*}