Question

How do you use implicit differentiation to find `(dy)/(dx)` given `2xy-y^2=3`?

Answer 1

`(dy)/(dx)= y/(y-1)`

Explanation:

Given
`color(white)("XXX")color(red)(2xy)-color(blue)(y^2) = color(green)3`

Note 1:
`color(white)("XXX")color(red)((d(2xy))/(dx)) = 2x * (dy)/(dx) + y * (d(2x))/(dx) =color(red)( 2(dy)/(dx)+2y)`

Note 2:
`color(white)("XXX")color(blue)((d(y^2))/(dx)) = color(blue)(2y(dy)/(dx))`

Note 3:
`color(white)("XXX")color(green)((d(3))/(dx))=color(green)(0)`

Therefore `2xy-y^2=3`

`rArr color(red)(2(dy)/(dx)+2y)-color(blue)(2y(dy)/(dx))=color(green)(0)`

`rArr (dy)/(dx)-y(dy)/(dx)= -y`

`rArr (dy)/(dx)(1-y) = -y`

`rArr (dy)/(dx)= y/(y-1)`