I know how to take derivatives I'm just not use to the notation y'all are using
Are you saying that you do not know how to find a derivative of a polynomial? If you are taking a course with questions about partial derivatives, that would be very strange!
A general polynomial in x and y would be of the form, as Fernando Revilla said, with only a finite number of non-zero.
Okay, what is ? What is ? Now differentiate with respect to y and with respect to x.