first lets get the derivatives with respect to #x# and #y#
#d/dx = y+(y(y-1))/(2sqrt(xy-x))#
#d/dx = (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))#
#d/dy = x+sqrt(xy-x) +xy(1/(2sqrt(xy-x)))#
#d/dy = (2xsqrt(xy-x))/(2sqrt(xy-x))+ (2(xy-x))/(2sqrt(xy-x)) +(xy)/(2sqrt(xy-x))#
#d/dy = (2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))#
next note that
#d/dx=d/dy⋅dy/dx# or equivalently #d/dy⋅dy/dx -d/dx=0#
which leads to
#(2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx) - (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))=0#
# (2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))#
# (dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x)) *(2sqrt(xy-x))/(2xsqrt(xy-x) + 2x(y-1)+(xy))#
# (dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2xsqrt(xy-x) + 2x(y-1)+(xy))#
# (dy)/(dx)=(y2sqrt(xy-x)+ y^2-y)/(2xsqrt(xy-x) + 3xy - 2x)#