If `y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+...oo))))`, then `(dy)/(dx)` is equal to

Options

A. `(y + x)/(y^(2) + x)`
B. `(y^(3) + x)/(2y^(2) - 2xy -1)`
C. `(y^(3) + x)/(2y^(2) -x)`
D. None of these

Answer

D

Explanation

We have
`y = sqrt(x + sqrt(y+ sqrt(x+ sqrt(y + ...oo))))`
`rArr y^(2) = x + sqrt(y+ sqrt(x + sqrt(y + ...oo)))`
` rArr y^(2) = x + sqrt(y + y)`
`rArr (y^(2) - x)^(2) = 2y `
On differentiating both sides w.r.t.w, we get
`2(y^(2) -x)(2y (dy)/(dx) -1) = 2 (dy)/(dx) `
` rArr (dy)/(dx) = (y^(2) - x)/(2y^(3) - 2xy -1)`