We start from the given equation
#(x/y-2)(xy-3)-e^y=C#
Expand the left side of the equation
#(x/y)(xy)-(3x)/y-2xy+6-e^y=C#
#x^2-(3x)/y-2xy+6-e^y=C#
Differentiate both sides of the equation with respect to #x#
#d/dx(x^2-(3x)/y-2xy+6-e^y)=d/dx(C)#
#d/dx(x^2)-3*d/dx(x/y)-2*d/dx(xy)+d/dx(6)-d/dx(e^y)=0#
#2x-3((y*1-x*y')/y^2)-2*(xy'+y*1)+0-e^y*y'=0#
Substitute right away the value of #x=-2# and #y=1# from the point #(-2,1)#
#2x-3((y*1-x*y')/y^2)-2*(xy'+y*1)+0-e^y*y'=0#
#2(-2)-3((1*1-(-2)*y')/1^2)-2[(-2)y'+1*1]+0-e^1*y'=0#
Simplify
#2(-2)-3((1*1-(-2)*y')/1^2)-2[(-2)y'+1(1)]+0-(e^1)y'=0#
#-4-3((1+2y')/1)+4y'-2+0-(e^1)y'=0#
#-4-3-6y'+4y'-2-ey'=0#
Solve for #y'#
#-6y'+4y'-ey'=4+3+2#
#(-6+4-e)y'=9#
#y'=9/(-6+4-e)#
#y'=9/(-2-e)#
#y'=(-9)/(2+e)#
#y'=-1.90747#
God bless....I hope the explanation is useful.
