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I was bothered by that one on the test as well. I definitely got it wrong on the test but looking at it afterward I'm pretty sure you were on track if you had some crazy gradient. The function was $(x^{2}-y^{2})e^{(-x^{2}+y^{2})}$. Looking at it after the test I'm pretty sure the gradient should come out to $$\langle2xe^{-x^{2}+y^{2}}(x^{2}-y^{2}-1),2ye^{-x^{2}+y^{2}}(-x^{2}+y^{2}+1) \rangle$$.

That answer was also confirmed by wolfram alpha so if you had some crazy algebra you were dealing with I imagine you were on the right track.