Question

How do you use implicit differentiation to find `(dy)/(dx)` given `3x^2y^2=4x^2-4xy`?

Answer 1

`dx/dy=(8x-4y-6xy^2)/(6x^2y + 4x)`

Explanation:

Given -

`3x^2y^2=4x^2-4xy`
`6xy^2+6x^2y.dx/dy=8x-4y+(-4x.dy/dx)`
`6xy^2+6x^2y.dx/dy=8x-4y-4x.dy/dx`

`6xy^2+6x^2y.dx/dy+4x.dy/dx=8x-4y`
`6x^2y.dx/dy+4x.dy/dx=8x-4y-6xy^2`
`(6x^2y + 4x)dx/dy=8x-4y-6xy^2`
`dx/dy=(8x-4y-6xy^2)/(6x^2y + 4x)`

Answer 2

`dy/dx = frac{4x-2y-3xy^2}{2x+3xy^2}`

Explanation:

`3x^2y^2 = 4x^2-4xy`

Product and power rule:
`3(2x)(y^2) + 3(x^2)(2y)(dy/dx) = 8x-4(y + x dy/dx)`

`6xy^2 + 6x^2y dy/dx = 8x-4y-4x dy/dx`

Move all terms that include `dy/dx` to one side:
`4x dy/dx + 6x^2y dy/dx = 8x-4y-6xy^2`

`dy/dx (2)(2x+3x^2y) = 2(4x-2y-3xy^2)`

`dy/dx = frac{4x-2y-3xy^2}{2x+3xy^2}`