calculus - Differentiate $:textrm{ln}sqrt{textrm{ln}:x}$

I am confused about the solution and method of differentiating this function:

$$\frac{d}{dx}\:\textrm{ln}\sqrt{\textrm{ln}\:x}$$

Why is ln not considered a constant and then multiplied by the derivative of$\:\sqrt{\textrm{ln}\:x}$ ?

The solution is given as:

$$\left(\frac{1}{2x\:\textrm{ln}x}\right)$$
How exactly is the chain rule applied to the entire function at once?