**Solution.**Let F(x) = \sec^2(x), f(u) = u^2 and g(x) = \sec(x). Then we will apply the chain rule:

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

\begin{align*} f'(u) = 2u \quad \text{and} \quad g'(x) = \tan(x)\sec(x). \end{align*}

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= 2\sec(x)\tan(x)\sec(x) \\ &= 2\tan(x)\sec^2(x). \end{align*}