Solution:

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

On derivativing both the sides with respect to $x$.

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

$=\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^{2} \sqrt{x} \cdot \frac{1}{2} x^{\frac{1}{2}-1}$

$=\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^{2} \sqrt{x} \cdot \frac{1}{2 \sqrt{x}}$