We need to differentiate
#e^sqrt(xy)-xy=6#
We see that there are three terms.
In the first term we use chain rule along-with product rule,
#d/dx(ucdotv)=ucdot (dv)/dx+vcdot (du)/dx#
for the second term product rule and
since the third term is a constant it derivative #=0#
#D(e^sqrt(xy))-D(xy)=D(6)#
The exponent of first term can be written as #(xy)^(1/2)#
#e^sqrt(xy)cdot D (xy)^(1/2)-D(xy)=0#
#e^sqrt(xy)cdot 1/2 cdot (xy)^(-1/2)cdotD(xy)-(xcdot y'+y)=0#
#e^sqrt(xy)cdot 1/2 cdot (xy)^(-1/2)cdot(xcdot y'+y)-(xcdot y'+y)=0#
#(e^sqrt(xy))/ (2 cdot sqrt(xy))cdot(xcdot y'+y)-(xcdot y'+y)=0#
Let #(e^sqrt(xy))/ (2 cdot sqrt(xy))=G(x,y)#
Above expression becomes
#Gcdot(xcdot y'+y)-(xcdot y'+y)=0#,
#Gcdotxcdot y'+Gcdot y-xcdot y'-y=0#, solving for #y'#
#(G-1)cdotxcdot y'+(G-1)cdoty=0#,
dividing both sides with #(G-1)#
#xcdot y'+y=0#
# y'=-y/x# with the condition #(G-1)!=0#
