If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+\ dotto\ oo)))` , prove that `(dy)/(dx)=(cosx)/(2y-1)`

Explanation

We have `=sqrt(sin x+sqrt(sin x + sqrt(sin x +...,)))`
`rArr y=sqrt(sin x + y)`
`rArr y^(2)= sin x + y`
Differentiating both sides w.r.t. x, we get
`2y(dy)/(dx)=cos x + (dy)/(dx)`
`rArr (dy)/(dx) = (cos x)/(2y-1)`
Alternative method :
From (1), `y^(2)-sin x -y =0`
`therefore (dy)/(dx) =- ("differentitation of f(x,y) w.r.t.x keeping y as constant")/("differenetiation of f(x,y) w.r.t. y keeping x as constant")`
`=-(-cos x)/(2y-1)=( cos x)/(2y-1)`