How do I find #d/dxln (tan (x^2))#?
I would use the Chain Rule.
In a way it is like those russian dolls called Matrioskas where you have a smaller doll inside another. To get to the last one you have to open the big ones first.
(Picture source: https://ghaiaesoterico.wordpress.com/2013/08/20/lenda-de-matrioska/)
Here the smallest doll is the function
To open in this case means to derive.
So you get: