Can I get some help on implicit differentiation?

Question

Here is the function: 2y^2 - 3x^2 = 18 - 2xy

Answer 1

Differentiation of 2y^2 with respect to y is 2y, but with respect to x is 2y dy/dx

Differentiating the given equation with respect to x,

4y dy/dx - 6x = -2[y + x dy/dx]

=> (4y + 2x) dy/dx = 6x - 2y

=> dy/dx = (3x -y) / (2y + x).

Answer 2

With implicit differentiation, you take derivatives dy/dx (wherever there is a y) and dx/dx (wherever there is an x).  You treat y's like it is a dependent variable, just like you'd use the chain rule.  For example:

if we took the derivative of:  (2x + 5)^2

we'd take the derivative of the outside function, then multiply that by the derivative of the inside function:

[2(2x + 5)]*(2) = 8x + 20

FOIL the original example (2x + 5)^2 = 4x^2 + 20x + 25

now do the derivative of that:  8x + 20 (same answer).

Now to your problem:

2y^2 - 3x^2 = 18 - 2xy

the 2y^2 part we'll do the chain rule trick:

2(2y)*(dy/dx)

2(2y)(dy/dx) - 6x(dx/dx) = - 2y(dx/dx) - 2x(dy/dx)

(note:  the -2xy part uses the product rule)

simplify (the dx/dx isn't necessary, since dx/dx=1)

4y(dy/dx) - 6x = - 2y - 2x(dy/dx)

now get all the dy/dx terms on 1 side:

4y(dy/dx) + 2x(dy/dx) = 6x-2y

(4y + 2x) (dy/dx) = 6x-2y

dy/dx = (6x-2y)/(4y+2x)

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