A function is differentiable at a point if it is continuous at that point, and if its partial derivatives are continuous at that point.
So to check continuity, we need to check that the limit exists and is equal to 0 at the point (x, y) = (0, 0). The easiest way is to convert to polars.
Clearly, this value will change depending on the value of , so the limit does not exist at that point.
Since the function is not continuous at that point, the function is not differentiable at that point.