Like an onion, you must peel back the layers, working from the outside in. That means chain rule first, with a product rule inside. On the RHS, is it (e^x)y or e^(xy)? You want to be explicit with parentheses.
For the left hand side may I ask Ackbeet: are you assuming that y is a function of x in order to use both the chain rule and product rule when differentiating wrt x?
I would have just use the chain rule and treat y as constant...? I mean having z=f(xy) doesnt mean that y is a function of x right?