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The math below was written in a thread I needed help in.

Define $$F(x,y,z)=x^2y+xz^2-5.$$

Hence $$\frac{\partial x}{\partial y}=-\frac{F’_y}{F’_x}=-\frac{x^2}{2xy+z^2}\,,$$$$\frac{\partial x}{\partial z}=-\frac{F’_z}{F’_x}=-\frac{2xz}{2xy+z^2}\,.$$

It seems more intuitive to me that $\frac{\partial x}{\partial y} = \frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial x}} = \frac{F'_y}{F'_x}$

I don't see where the negative sign comes from!

slifer227
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  • In $\partial x \partial y$ and $\partial x/\partial z$, note that $x$ is a function of two variables $y$ and $z$. But in $F'_x$, etc., note that $F$ is a function of three variables $x,y,z$. – GEdgar Nov 24 '19 at 23:31
  • See https://math.stackexchange.com/questions/26205/what-is-the-implicit-function-theorem, https://math.stackexchange.com/questions/1006492/proof-of-multivariable-implicit-differentiation-formula/2575261, and similar questions. – Hans Lundmark Nov 25 '19 at 08:31

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