Thread: Isolating a Variable in Multivariable Function

1. 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.

2. 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:
$4x^3+ 0+ 4z^3\frac{\partial z}{\partial x}= 4yz+ 4xy\frac{\partial z}{\partial x}$
Now solve that linear equation for $\frac{\partial z}{\partial x}$.