How to multiply partial derivatives?

Question

If x, y and z are three quantities which satisfy a functional relationship f(x, y, z) = 0

show that

( ∂y/∂x)( ∂x/∂y ) = 1

I think I need to use the chain rule somewhere in there, not sure

Answer 1

The rule is $$\left(\frac{\partial a}{\partial b}\right)_c\left(\frac{\partial b}{\partial d}\right)_c=\left(\frac{\partial a}{\partial d}\right)_c,$$where all partial derivatives are defined with the same subscripted variable(s) $c$ fixed. In your case, $a=d=y,\,b=x,\,c=z$, so the product simplifies to $\left(\frac{\partial y}{\partial_y}\right)_c=1$ as required. Note that in the triple product rule @lulu linked to, the susbcripted fixed variables vary among the partial derivatives, so the above chain rule is unavailable. There is thus no contradiction with the TPR obtaining $\color{blue}{-}1$.