How do you prove the partial derivative identity with three variables?

[mj1357]

Homework Statement

Suppose that the equation f(x,y,z)=0 can be solved for each of the three variables as a differentiable function of the other two. Prove that:

(dx/dy)(dy/dz)(dz/dx)=-1

Homework Equations

The Attempt at a Solution

In the case of two variables where one is a function of the other, dy/dx = -(df/dx)/(df/dy), but I can't figure out how this works with 3 variables.

 

[lanedance]

so to get you started, first we know
x=x(y,z)
y=y(z,x)
z=z(x,y)


so fa(x,y,z) = f(x,y,z(x,y))=0 the differentiating w.r.t x gives

df/dx +df/dz(dz/dx)=0

 

[mj1357]

Thanks. I was able to figure it out from there.