How Does the Chain Rule Relate ∂h/∂z to ∂g/∂x?

[cruxcriticoru]

If given an implicit function f(x,y,z)=0. Then, we can get z=g(x,y) and x= h(y,z).
I know the answer for the partial derivative of g(x,y)' for x, how can I know the partial derivative of h(y,z) for z?

I know f( x, y, z)=0. And I know ∂g / ∂x is positive.
How can I define whether ∂h / ∂z is negative or positive?
How can I express ∂h / ∂z in terms of ∂g / ∂x?

 

[CompuChip]

Hi cruxcriticoru, welcome to PF.

Probably your answer involves the chain rule. You know that ∂g / ∂x = ∂z / ∂x is positive. Can you relate ∂h / ∂z with ∂z / ∂x using the chain rule?