Question

How do I find `d/dxln (tan (x^2))`?

Answer 1

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 `x^2` but to get to it you have first to "open" `tan` and `ln`.

To open in this case means to derive.

So you get:

`d/dxln(tan(x^2))=1/(tan(x^2))*1/(cos^2(x^2))*2x`

Remember that:
`d/dxln(x)=1/x`
`d/dxtan(x)=1/(cos^2(x))`
`d/dxx^2=2x`