It looks almost perfect. It sounds like you know what's going on. I didn't check you calculation of , but this is not the hardest part anyway.
you forget to apply to (1,0) (so that instead of e.g. , you want ).
As for a faster way of solving this, I guess you could save a wee bit of time, but it's not a method I would utilize in a small problem like this. Specifically, you are being asked only for . For calculating this, you do not need the second column of the matrix for , so you could save yourself some work by calculating only the first column.
In this problem, the partial derivatives of are being taken with respect to the variables and . Since and are, in turn, functions of the variables , you could also consider as a function of and . But the partial derivatives of with respect to and will be completely different from the partial derivatives of with respect to and .