u=(e^x)cos(xy)
using the product rule,
u_x = e^x.cos(xy) - y.e^x.sin(xy)
u_xy = -x.e^x.sin(xy) - (e^x.sin(xy) + xy.e^x.cos(xy))
u_xy = e^x.sin(xy)(-x - 1) - xy.e^x.cos(xy)
u_y = -x.e^x.sin(xy)
u_yx = -e^x.sin(xy) - x(e^x.sin(xy))'
u_yx = -e^x.sin(xy) - x(e^x.sin(xy) + y.e^x.cos(xy))
u_yx = e^x.sin(xy)(-1 - x) - xy.e^x.cos(xy)
Therefore: d2u/dx/dy = d2u/dydx