Thread: Isolating a Variable in Multivariable Function

This is actually for a Calculus question, but the Algebra part is what I'm stumped on.

I'm given the equation:
x^4 + y^4 + z^4 = 4xyz

And I will need to find the partial derivatives of z with respect to x and y. That part I can do. But I haven't been able to isolate z.

I feel like this should be relatively simple, but I'm missing something.

Originally Posted by TechVick

This is actually for a Calculus question, but the Algebra part is what I'm stumped on.

I'm given the equation:
x^4 + y^4 + z^4 = 4xyz

And I will need to find the partial derivatives of z with respect to x and y. That part I can do. But I haven't been able to isolate z.

I feel like this should be relatively simple, but I'm missing something.

It is not easy to isolate z. That would mean solving a fourth degree equation.

However, if you just want to find the derivative of z with respect to x, you don't need to solve for z. Use "implicit differentiation"- differentiate each part of the equation with respect to x, writing [tex]\frac{\partial z}{\partial x} for the unknown derivative:

Now solve that linear equation for .