This is the content of differential calculus of multivariate functions in higher mathematics, and dz is total differential.

dz=z'(x)dx+z'(y)dy

=e^[-(x+y)^2][ycosxy-2(x+y)sinxy]dx+e^[-(x+y)^2][xcosxy-2(x+y)sinxy]dy

(Maybe. The process is so simple and the answer is long, so it's not surprising)