Identify the layers of the function for the chain rule.

The function is $y = \sec^2(\sqrt{\sin(x)})$. This can be written as $y = (\sec(\sqrt{\sin(x)}))^2$.
Let $u = \sec(\sqrt{\sin(x)})$. Then $y = u^2$.
Let $v = \sqrt{\sin(x)}$. Then $u = \sec(v)$.
Let $w = \sin(x)$. Then $v = \sqrt{w} = w^{1/2}$.

Differentiate $y = u^2$ with respect to $u$:

Differentiate $u = \sec(v)$ with respect to $v$:

Differentiate $v = w^{1/2}$ with respect to $w$:

Differentiate $w = \sin(x)$ with respect to $x$:

Substitute back the expressions for $u, v, w$:

The final answer is $\boxed{\frac{\sec^2(\sqrt{\sin(x)}) \tan(\sqrt{\sin(x)}) \cos(x)}{\sqrt{\sin(x)}}}$.