Maths project

The Project
Frank G. is teaching AP Calculus. In his class, a few students completed the problem different ways. He sends the following message to an online message board for AP Calculus teachers for assistance.
OK people I need your help on this one. What are your thoughts on this?
For the equation x^2=( x+y )/(x-y), find ( dy )/dx.
I tried to solve the problem implicitly using the quotient rule, then multiplied by (x-y) and used the product rule, and then solved for y and did it explicitly.
Three different answers! How can this be?
Your thoughts, please.
Frank G.
Your job is to help this teacher out.
Find the 3 different answers that Frank G did using the 3 methods he described. Be sure to label the ways you are using.
Show that your 3 different answers are all equal. HINT: You will need to solve for y and plug this into your first two answers.
Write a message to Frank stating your answers for each problem to verify you both got the same results and then tell him how they are all equal.

Sample Solution