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#