Here ,
sqrt(xy)=x-y/(x-1)√xy=x−yx−1
Diff. each term w.r.t. xx, (using product,quotient and chain rules)
1/(2sqrt(xy))*d/(dx)(xy)=1-((x-1)y_1-y(1))/(x-1)^212√xy⋅ddx(xy)=1−(x−1)y1−y(1)(x−1)2
=>1/(2sqrt(xy))[xy_1+y*1]=1-y_1/(x-1)+y/(x-1)^2⇒12√xy[xy1+y⋅1]=1−y1x−1+y(x−1)2
=>(xy_1)/(2sqrt(xy))+y/(2sqrt(xy))=1-y_1/(x-1)+y/(x-1)^2⇒xy12√xy+y2√xy=1−y1x−1+y(x−1)2
=>(xy_1)/(2sqrt(xy))+y_1/(x-1)=1+y/(x-1)^2-y/(2sqrt(xy))⇒xy12√xy+y1x−1=1+y(x−1)2−y2√xy
y_1{x/(2sqrt(xy))+1/(x-1)}=(2sqrt(xy)(x-1)^2+2ysqrt(xy)-y(x-
1)^2)/(2sqrt(xy)(x-1)^2)y1{x2√xy+1x−1}=2√xy(x−1)2+2y√xy−y(x−1)22√xy(x−1)2
y_1{(x^2-x+2sqrt(xy))/(2sqrt(xy)(x-1))}y1{x2−x+2√xy2√xy(x−1)}=(2sqrt(xy)(x-
1)^2+2ysqrt(xy)-y(x-1)^2)/(2sqrt(xy)(x-1)^2)2√xy(x−1)2+2y√xy−y(x−1)22√xy(x−1)2
y_1(x^2-x+2sqrt(xy))=(2sqrt(xy)(x-1)^2+2ysqrt(xy)-y(x-1)^2)/((x-1))y1(x2−x+2√xy)=2√xy(x−1)2+2y√xy−y(x−1)2(x−1)
:.y_1=(2sqrt(xy)(x-1)^2+2ysqrt(xy)-y(x-1)^2)/((x^2-x+2sqrt(xy))(x-1))
Where, y_1=(dy)/(dx)
