Order of Divisor Function $d(n)$

Question

Let $d(n)$ be number of positive divisor of natural number $n$. It is known from $d(n)=O(n^{\varepsilon})$ for every $\varepsilon>0$, we can deduce that $d(n)=n^{C/(\log \log n)}$ for some constant $C$. I am curious whether there is a proof for this fact directly and deduce $d(n)=O(n^{\varepsilon})$ as a corollary. Does anybody know how to do this or have reference to the proof?