Solving the Chain Rule Equation with Differential Calculus

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Homework Statement

Homework Equations

The Attempt at a Solution

a) (∂z/∂x)=-(∂f/∂x)*(∂z/∂f)

i used that (AxB)=-(BxA)
so i get

(∂z/∂x)=-[-(∂z/∂f)(∂f/∂x)]
=(∂z/∂x)
is this correct if not can someone give me hints pls

thanks

 

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i just realized that i can not use that cross product...now I am really lost help...someone pls give me a hint...pls

 

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anyone? pls

 

[HallsofIvy]

Let f(x, y)= F(x, y, z(x,y))= 0 so that f is a constant function and all its partial derivatives are 0. Then
$$\frac{\partial f}{\partial x}= \frac{\partial F}{\partial x}+ \frac{\partial F}{\partial z}\frac{\partial z}{\partial x}= 0$$.

Solve that for
$$\frac{\partial z}{\partial x}$$.