Question

How do you differentiate `xy^2-x^3y=6`?

Answer 1

You have an Implicit Function in which `y` is function of `x` so you have to derive it as well.

You get:
`(1*y^2+2xydy/dx)-(3x^2y+x^3dy/dx)=0` using the Product Rule.

Then:
`y^2+2xydy/dx-3x^2y-x^3dy/dx=0`

Collecting `dy/dx`:
`dy/dx[2xy-x^3]=3x^2y-y^2`
`dy/dx=(y(3x^2-y))/(x(2y-x^2))`

Hope it helps