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

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SUMMARY

The discussion centers on the application of the chain rule in multivariable calculus, specifically relating the partial derivatives ∂h/∂z and ∂g/∂x. It establishes that given an implicit function f(x,y,z)=0, the relationship between these derivatives can be expressed through the chain rule. The participants confirm that since ∂g/∂x is positive, the sign of ∂h/∂z can be determined by analyzing the relationship between ∂z/∂x and ∂h/∂z. This establishes a clear mathematical connection between the derivatives involved.

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cruxcriticoru
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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?
 
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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?
 

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