Consider the curve defined by 2x2+3y2-4xy=36 2 x 2 + 3 y 2 - 4 x y = 36.

(a) Show that dy/dx=2y-2x / 3y-2x

Step-by-step explanation:

Given a curve defined by the function 2x²+3y²-4xy=36

The total differential of this function with respect to a variable x makes the function an implicit function because it contains two variables.

Differentiating both sides of the equation with respect to x we have:

4x+6ydy/dx - (4xd (y) / dx+{d (4x) / dx (y)) } = 0

4x + 6ydy/dx - (4xdy/dx + 4y) = 0

4x + 6ydy/dx - 4xdy/dx - 4y = 0

Collecting like terms

4x-4y+6ydy/dx - 4xdy/dx = 0

4x-4y + (6y-4x) dy/dx = 0

4x-4y = - (6y-4x) dy/dx

4y-4x = (6y-4x) dy/dx

dy/dx = (4y-4x) / 6y-4x

dy/dx = 2 (2y-2x) / 2 (3y-2x)

dy/dx = 2y-2x/3y-2x proved!