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Suppose that $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is such that $f(x,y,z) = 0$ for some $x,y,z$. Then, I am asked to show that, if each variable $x,y,z$ can be defined as an implicit function of the other ones, then $\frac{dx}{dy} \frac{dy}{dz} \frac{dz}{dx} = -1$. I tried to use the Implicit Function Theorem, but I failed at trying to obtain this equality. I would be grateful if someone could give me just a hint, not the entire answer since I would like to conclude it by myself unless I say the opposite.

Daniel C.
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  • duplicate https://math.stackexchange.com/questions/392770/proof-on-showing-if-fx-y-z-0-then-product-of-partial-derivatives-evaluated-at or https://math.stackexchange.com/questions/942457/understanding-frac-partial-x-partial-y-frac-partial-y-partial-z-frac – Levon Minasian Nov 24 '23 at 11:41
  • If one solves for $x$, it is a function of $y$ and $z$, not of $y$ alone. Thus the correct notation here is $$\dfrac{\partial x}{\partial y}\dfrac{\partial y}{\partial z}\dfrac{\partial z}{\partial x} = -1$$ and the partial derivatives are each understood as holding the third variable constant. As for the hint, if you have not already read Levon Minasian's links, use implicit differentiation on $f(x,y,z) = 0$ to express each of the partial derivatives in terms of the three partial derivatives $f_x, f_y, f_z$, – Paul Sinclair Nov 25 '23 at 16:44
  • Thanks for your explanation, I have finally proved this statement. – Daniel C. Nov 26 '23 at 10:13

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