यदि `x = y+(1)/(y)""_(+(1)/(y)""_(+(1)/(y)""_(+..."to"oo)))` , सिद्ध करें कि (prove that)
`(dy)/(dx)=2x^(2)+y^(2)-3xy`

Explanation

दिया है `x = y+(1)/(y)""_(+(1)/(y)""_(+(1)/(y)""_(+..."to"oo)))`
`thereforex=y+(1)/(x)` ... (1)
x के सापेक्ष अवकलित करने पर,
`(dy)/(dx)-(1)/(x^(2)),` या `(dy)/(dx)=1+(1)/(x^(2))`
या `(dy)/(dx)=1+(x-y)^(2), " " [because(1), (1)/(x)=x-y]`
`=1+x^(2)+y^(2)-2xy` ... (2)
(1) से, `x^(2)=xy+1 " " therefore1=x^(2)-xy`
(2) में रखने पर हमें मिलता है, `(dy)/(dx)=x^(2)-xy+x^(2)+y^(2)-2xy`
अतः `(dy)/(dx)=2x^(2)+y^(2)-3xy`