Implicitly differentiating PDE (multivariable calculus)

[Legion81]

The problem:
Find the value of dz/dx at the point (1,1,1) if the equation xy+z³x-2yz=0 defines z as a function of the two independent variables x and y and the partial derivative exists.

I don't know how to approach the z³x part. I thought you would use the product rule and get 3(dz/dx)²x + z³. But if that is right, the final equation looks something like

y + 3x(dz/dx)² + z³ - 2y(dz/dx) = 0

And I don't think that is right. The only way I know to solve that would be with the quadratic equation and that gives a complex value. Am I forgeting the chain rule somewhere or just way off on approaching this problem?

 

[Frillth]

I have done very little multivariable so this could easily be wrong, but if you implicitly differentiated z^3x shouldn't you get 3z^2x*dz/dx + z^3?

 

[Legion81]

I have done very little multivariable so this could easily be wrong, but if you implicitly differentiated z^3x shouldn't you get 3z^2x*dz/dx + z^3?

That's right because you have the 3z²x(dz/dx) + z³(dx/dx). I don't know what I was thinking... Thank You!

 

[ducnguyen2000]

You didn't apply the chain rule, remember df(u)/dx = f(u)'*du/dx