[Did Jacob take Joseph’s coat with him to Egypt or leave it behind in the land of Canaan?]
It was destroyed.
After his brothers sold JOSEPH into slavery, they decided deceive their father. They killed one of the goats, and dipped Joseph’s coat in its blood; (and probably tore the coat) and they brought it to their father, and they said to him: “We found this coat out in the wilderness. Look at it, father, and tell us if you think it was the coat of your son.”
And Jacob knew it at once. He said: “It is my son’s coat. Some wild beast has eaten him. There is no doubt that Joseph has been torn in pieces!”
http://biblehub.com/library/marshall/the_wonder_book_of_bible_stories/the_story_of_joseph_and.htm
Yes, the brothers dipped the coat in goat’s blood, but it doesn’t say they tore it. Jacob rent his clothes when he saw Joseph’s coat, but it doesn’t say he did anything to the coat of many colors.