Let $f:\mathbb{R}^{3} \to \mathbb{R}$ given by $f(x,y,z) = x^{2}yz$. Prove that in a neighborhood of $(1,1,1)$, the equation $f(x,y,z) = 1$ define $x$ as function of class $C^{\infty}$ of the variables $y$ and $z$, and obtain the partial derivatives of this function. Calculate, explicitly, this function and check the results obtained.
The first part is direct consequence of the Implicit Function Theorem. I'm having trouble getting the function explicitly. Can someone help me?
Near $(1,1,1)$, $y$ and $z$ are greater than $0$. Therefore, you can define $x$ as a function of $y$ and $z$ by$$x=\frac1{\sqrt{yz}}.$$It is clear then that $x^2yz=1$.