Question

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

Answer 1

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)`