Expert: Steve Holleran - 10/4/2007

Question
I am having a really hard time trying to solve the equation x^3+y^3=18xy. It is an implicit differentaion problem.

Answer
Hi Jen,

Okay,  by "solving" I assume that you are trying to differentiate it to find dy/dx, is that right?

If so, then here's what you want to do:  apply the normal derivative rules as you go through the equation, except that whenever you are differentiating a "y" term, tag on a "dy/dx" to it:

 (On the right side, we need to use the Product Rule)

              d/dx[ x^3 + y^3 = 18xy ]

       3x^2 + 3y^2 * dy/dx = 18 * [ x * dy/dx + y * 1]

       3x^2 + 3y^2 * dy/dx = 18x * dy/dx + 18y

              3y^2 * dy/dx - 18x * dy/dx = 18 y - 3x^2

            dy/dx * (3y^2 - 18x) = 18y - 3x^2

so then      dy/dx = (18y - 3x^2) / (3y^2 - 18x)

or, reducing by 3 on the right:

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

I hope this is what you needed.

Steve