1. State the definition of the partial derivative of a function f(x,y) with respect to y.

2. Find the value of  if it exists, or show that it does not exist.

As usual, we try approaching along several different paths:

First let's try along the line x=0, so we have .

Okay, let's try along the line y=0, so we have .

So far they agree, but let's try a third, like y=x, where we have .
 

Okay, so getting that far and saying something like "So they all agree and I think the actual limit is 0" is all I was looking for. If you want somethng more convincing, the best way to go is a little algebra like . That's nice and conclusive, but given the mix-up in the question I asked I didn't require that you do anything like it for full credit (but congratulations to the people who did!).