Thread: constrained partial derivative identity

if $\displaystyle f(x,y,z)=0$ then,
$\displaystyle (\frac{\partial x}{\partial y})_z (\frac{\partial y}{\partial z})_x (\frac{\partial z}{\partial x})_y = -1 $

I tried expressing $\displaystyle \frac{\partial f}{\partial x}$ with z constant but this didnt work

Im lost. Help much appreciated

It should! With z constant and x a function of y, we can think of f as a function of y only:
$\displaystyle \frac{df}{dy}= \frac{\partial f}{\partial x}\frac{\partial x}{\partial y}+ \frac{\partial f}{\partial y}= 0$
so
$\displaystyle \left(\frac{\partial x}{\partial y}\right)_z= -\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}$

Now do the same for the other two

Why have you written $\displaystyle \frac{df}{dy}$ should it not be $\displaystyle \frac{\partial f}{\partial y}$ as $\displaystyle z=g(x,y)$

This is where my confusion arises