यदि ` xsqrt( (1+y) )+ ysqrt((1+ x))=0` तब सिद्ध कीजिए की `(dy)/(dx) =- (1+x)^(-2)`

Explanation

` " "xsqrt ( (1+y))+ysqrt((1+x))=0 `
` rArr" "xsqrt (( 1+y) )=-ysqrt((1+x))`
` rArr " "x^(2) (1+y)=y^(2) (1=x) `
` " " `(दोनों और का वर्ग करने पर )
` rArr" "(x^(2) -y^(2))+xy(x-y) =0 `
` rArr " "(x-y) [x+ y+ xy] =0`
`rArr x-y=0 ` या ` " " x+y (1+x) =0 `
यदि ` x- y=0 " "rArr " "y=x " "rArr " "(dy)/(dx) =1 `
yeh अभीष्ट परिणाम है
अतः ` " "x+y (1+x) =0 " "rArr y= (-x) (1+x)`
`therefore (dy)/(dx) =- [((1+x) (d)/(dx) (x) -x(d)/(dx) (1+x))/((1+x)^(2))]`
` " "=-[ ((1+ x)-x) /((1+x )^(2))=- [(1)/((1+x)^(2))]`
` therefore " "(dy)/(dx) =-(1+x)^(2) `