finding a partial derivative

1. The problem statement, all variables and given/known data

Given f(x, y, z) = 0, find the formula for



[tex]

(\frac{\partial y}{\partial x})_z

[/tex]





2. Relevant equations

Given a function f(x, y, z), the differential of f is

[tex]

df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz

[/tex]



3. The attempt at a solution



We know that f(x, y, z) = 0 so using above, I get

[tex]

df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz

= 0

[/tex]



We also know that we are finding the partial with constant z so I set dz = 0. I then divided by dx throughout and solve for [itex] \frac{\partial y}{\partial x} [/itex].



My final answer is

[tex]

(\frac{\partial y}{\partial x})_z = -\frac{\frac{\partial f}{\partial x} }{\frac{\partial f}{\partial y} }

[/tex]



I just wanted to confirm that I'm doing things correctly in finding this partial derivative.



Thanks!