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

2 Answers
Aug 15, 2017

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)

Aug 15, 2017

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}