Proving that $X\to aX|Y$, $Y \to Yab|b$ is unambiguous
First, figure out what the language is.
Then, try to do a few examples of productions in the grammar. What do you notice about the derivation trees and sequences? How would this help you to prove this grammar is unambiguous?
Hint: At each step in the process of deriving a word, what are the possible derivation rules you can use? How would each of them affect the resulting word?