Begin by verifying that the point #(1/2, 2)# is on the curve:
#e^(1/2(2))-2^2=e-4#
#e - 4 = e - 4 larr# verified.
Compute #dy/dx# using implicit differentiation:
#(d(e^(xy)))/dx-(d(y^2))/dx=(d(e-4))/dx#
For the first term, let #u = xy# and use the chain rule #(d(e^(xy)))/dx = (d(e^u))/(du) (du)/dx#
Use #(d(e^u))/(du) = e^u = e^(xy)#
Use the product rule for #(du)/dx = dx/dxy+xdy/dx = y + xdy/dx#
Returning to the equation:
#ye^(xy) + xe^(xy)dy/dx-(d(y^2))/dx=(d(e-4))/dx#
Use the chain rule for #(d(y^2))/dx = (d(y^2))/dydy/dx = 2ydy/dx#:
#ye^(xy) + xe^(xy)dy/dx-2ydy/dx=(d(e-4))/dx#
The derivative of a constant is 0:
#ye^(xy) + xe^(xy)dy/dx-2ydy/dx=0#
Collect all of the terms that do not contain #dy/dx# on the right:
#xe^(xy)dy/dx-2ydy/dx=-ye^(xy)#
Factor out #dy/dx#
#(xe^(xy)-2y)dy/dx=-ye^(xy)#
Divide both sides by the leading factor:
#dy/dx=-(ye^(xy))/(xe^(xy)-2y)#
Evaluate at the point #(1/2,2)#:
#dy/dx=-(2e^(1/2(2)))/(1/2e^(1/2(2))-2(2))#
#dy/dx = (4e)/(8-e) larr# selection C
