#sqrt(e^(2x-2y)) = (e^(2x-2y))^(1/2) = e^((1/2)(2x-2y)) = e^(x-y)#

Rewrite:

#sqrt(e^(2x-2y))-xy^2=6# as

#e^(x-y) - xy^2=6#.

Now differentiate term by term. Using the chain rule to differentiate #e^(x-y)# and the product rule for #-xy^2#

#d/dx(e^(x-y)) - d/dx(xy^2) = d/dx(6)#.

#e^(x-y)d/dx(x-y) - [y^2+x2ydy/dx] = 0#.

#e^(x-y)(1-dy/dx) - [y^2 + 2xydy/dx] = 0#.

Now solve for #dy/dx#. Distribute to remove brackets.

#e^(x-y) - e^(x-y)dy/dx - y^2 - 2xydy/dx = 0#.

#e^(x-y) - y^2 =e^(x-y)dy/dx + 2xydy/dx = 0#.

#e^(x-y) - y^2 =(e^(x-y) + 2xy)dy/dx = 0#.

#dy/dx = (e^(x-y) - y^2)/(e^(x-y) + 2xy)#