Question

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

Answer 1

Answer:

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!