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If y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+...oo))...

If `y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+...oo))))` then prove that `(dy)/(dx)=(y^2-x)/(2y^3-2xy-1)`

Text Solution

Verified by Experts

`y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+))))...oo=sqrt(x+sqrt(y+y))`
`"or "y^(2)=x + sqrt(2y)`
Differentiating w.r.t. x, we get
`"or "2ycdot(dy)/(dx)=1+(1)/(sqrt(2y))xx(dy)/(dx)`
`"or "(dy)/(dx)[2y-(1)/(sqrt(2y))]=1`
`"or "(dy)/(dx)=(sqrt(2y))/(2ysqrt(2y-1))=(y^(2)-x)/(2y^(3)-2xy-1)`
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