When you are constructing the Taylor polynomial (of any degree) you need to construct it around a point. Have you been given one such point? Usually, for hand calculations, you want to pick a point that is relatively close to the point you are trying to calculate. At the same time you want the Taylor polynomial to be easy to calculate. Have you been given a point to construct around? If not, you need to find the value of \arctan(1/3), which corresponds to x = 1 and y = 3. You cannot center the polynomial around this point, so you have to center it at a point that is close by and makes calculation easy.
The partial derivatives of this function are just fractions, so it doesn't really matter which point you choose. Which point did you choose? There are two that may interest you: (0,1) and (1,2) which are both the same distance. The first one has the added advantage that you are setting all x's to 0 which makes the form really simple. Furthermore, for the degree 0 term you will have to calculate arctan(0/3) which is just 0.
Of course, if you are more concerned about the accuracy you can just pick the point (1.98,3.02) as your center, but that will require you to calculate arctan(1.98/3.02) to a good precision for your constant term, which defeats the pupose. Now, commenting your work really means to just compare the answer you got to the actual answer and maybe say what percentage it is off by.