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}#