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If y=x+1/(x+1/(x+1/(x+\ dot))) , prove ...

If `y=x+1/(x+1/(x+1/(x+\ dot)))` , prove that `(dy)/(dx)=y/(2y-x)` .

Text Solution

Verified by Experts

We have
`y=x + (1)/(x + (1)/(x + (1)/(x+....)))=x+(1)/(y)`
`or y^(2)=xy+1`
` or 2y(dy)/(dx)=y+x(dy)/(dx)+0" [Differentiating both sides w.r.t.x]"`
`or (dy)/(dx)(2y-x)=y`
`or (dy)/(dx)=(y)/(2y-x)`
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