Which degree does the Taylor polynomial needs to have?
It's in general not easy to find the Taylor expansion of the composition of two function. There exists a formula, due to Faà di Bruno.
You can always do it the long way. Let .
We know that , so let and we find .
Differentiate both sides and we find
We know , so let and we find .
Differentiate both sides and we have
We know that so let and we find .
Follow this process to find as many terms as you need. I doubt that there will be a pattern.