Question

What is the implicit derivative of `xy+ysqrt(xy-x) =8`?

Answer 1

`(dy)/(dx)=(y2sqrt(xy-x)+ y^2-y)/(2xsqrt(xy-x) + 3xy - 2x)`

Explanation:

first lets get the derivatives with respect to `x` and `y`
`d/dx = y+(y(y-1))/(2sqrt(xy-x))`

`d/dx = (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))`

`d/dy = x+sqrt(xy-x) +xy(1/(2sqrt(xy-x)))`

`d/dy = (2xsqrt(xy-x))/(2sqrt(xy-x))+ (2(xy-x))/(2sqrt(xy-x)) +(xy)/(2sqrt(xy-x))`

`d/dy = (2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))`

next note that
`d/dx=d/dy⋅dy/dx` or equivalently `d/dy⋅dy/dx -d/dx=0`

which leads to

`(2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx) - (y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))=0`

`(2xsqrt(xy-x)+ 2x(y-1)+(xy))/(2sqrt(xy-x))(dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x))`

`(dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2sqrt(xy-x)) *(2sqrt(xy-x))/(2xsqrt(xy-x) + 2x(y-1)+(xy))`

`(dy)/(dx)=(y2sqrt(xy-x)+ y(y-1))/(2xsqrt(xy-x) + 2x(y-1)+(xy))`

`(dy)/(dx)=(y2sqrt(xy-x)+ y^2-y)/(2xsqrt(xy-x) + 3xy - 2x)`