How to prove the chain rule with respect to weak derivatives?

Question

The following theorem is on the textbook "weak differentialble functions". I found it confusing from the absolutely continuous part. I am writing to ask is this the only way to prove it? Can anyone give me some references to proofs in other books? Thanks so much!

Answer 1

You can use approx to proof chain. i.e., when you have a Sobolev function $u$, you can always build a smooth sequence $u_n$ such that $u_n\to u$ in $W^{1,p}$, and then work on $u_n$ instead $u$ first and finally push to the limit.

For a good proof, I would recommend you to read this book, page 129, theorem 4