I have tried to do this problem by brute force (just bloody calculating the mixed derivatives) and I get that both partial derivatives are equal at the origin. I find that

(Incase that didn't come out, I have also attached a pdf.)

What worries me is that I also used Mathematica to check this, and the result is still the same irrespective of the order of differentiation.

So, does anybody know an easier method of doing this, and what is going wrong? Does it have something to do with limits, as we need to compare at the origin? Thanks!

Sebastian