As far as I understand aratanaru and atarashii roughly translate to "new" . When each one have to be used or what are the differences?
Atarashii（新しい）, Aratana（新たな）and Aratanaru（新たなる）are all translated to "new". But, 新たなる is an exaggerated version of 新たな. In general, you don't use 新たなる in daily conversations.
It could be appeared in below contexts;
Last night's his live performance marked a new chapter in the history of Hip Hop. 新たなる1ページを刻んだ。
As a company slogan, "新たなる挑戦" as in a new challenge