Question

What is the implicit derivative of `xy+y/sqrt(xy-x) =2`?

Answer 1

`y'=(-y+2x*sqrt(xy-x)+y^2-2xy^2*sqrt(xy-x))/(-2x-2x^2sqrt(xy-x)+xy+2x^2ysqrt(xy-x))`

Explanation:

We have

`y+xy'+(y'sqrt(xy-x)-y*1/2*(xy-x)^(-1/2)(y+xy'-1))/(xy-x)=0`
multiplying by `sqrt(xy+x)*(xy-x)` we get

`(y+xy')(xy-x)sqrt(xy-x)+y'(xy-x)-y/2(y+xy'-1)=0`
multiplying by `2`

`2(y+xy')(xy-x)sqrt(xy-x)+2y'(xy-x)-y(y+xy'-1)=0`
`y'(-2x-2x^2sqrt(xy-x)+xy+2x^2ysqrt(xy-x))=-y+2xsqrt(xy-x)+y^2-2xy^2sqrt(xy-x)`

so we get

`y'=(-y+2xsqrt(xy-x)+y^2-2xy^2sqrt(xy-x))/(-2x-2x^2sqrt(xy-x)+xy+2x^2ysqrt(xy-x))`