I'm not sure how to solve this type of problem. Any help would be appreciated. Thanks.
You just need to apply the chain rule. The general formula for
$\displaystyle \frac{d}{dx}Arctan(u)=\frac{u'}{1+u^2}$
Where $\displaystyle u$ is a function of x. You're basically applying the chain rule here. Just remember that the derivative of $\displaystyle u=\sqrt{x}$ is $\displaystyle u'=\frac{1}{2\sqrt{x}}$, and $\displaystyle u^2=x$. Once you put it all together, you should obtain:
$\displaystyle \frac{1}{2\sqrt{x}(1+x)}$