The derivative of $\sec (\tan \sqrt{x})$ with respect to x is :

$\frac{\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^2 \sqrt{x}}{2 \sqrt{x}}$

$y = \sec (\tan \sqrt{x})$

differentiating wrt x

$\frac{d y}{d x} =\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \frac{d}{d x}(\tan \sqrt{x})$

$=\frac{\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^2 \sqrt{x} \frac{d}{d x} \sqrt{x}}{\frac{d y}{d x}}$         [Using chain rule]

$\frac{d y}{d x}=\frac{\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^2 \sqrt{x}}{2 \sqrt{x}}$