If `x ylog(x+y)=1,p rov et h a t(dy)/(dx)=-(y(x^2y+x+y))/(x(x y^2+x+y))`

Explanation

Given,`x ylog(x+y)=1`

Differentiating it with respect to x using the chain rule and product rule,

`(d/dx)(x ylog(x+y))=0`

=`xy(d/dx)(log(x+y))+xlog(x+y)(dy/dx)+ylog(x+y)(dx/dx)=0`

=`(xy)/(x+y)(1+(dy/dx))+xlog(x+y)(dy/dx)+ylog(x+y)=0`

=`(xy)/(x+y)+(xy)/(x+y)(dy/dx)+x(1/(xy))(dy/dx)+ylog(x+y)=0`

=`(dy/dx)((xy)/(x+y)+1/y)=-((xy)/(x+y)+1/x)`

=`(dy/dx)((xy^2+x+y)/(xy+y^2))=((xy^2+x+y)/(x^2+xy))`

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