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Finding a partial derivative

  1. Aug 2, 2014 #1
    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!
     
  2. jcsd
  3. Aug 2, 2014 #2

    maajdl

    User Avatar
    Gold Member

    You are right.
    I am just a little bit puzzled about the notations.
    How to you switch from the ratio dy/dx to the partial derivative.
    This should maybe be explained more explicitly.
     
  4. Aug 2, 2014 #3
    Hey, thanks for the quick response!

    That is a good point about the notation. Do you have any ideas on this? I don't have any good mathematical sense as to why I changed it from dy/dx to a partial derivative.
     
  5. Aug 2, 2014 #4
    You are correct. As was alluded to in your other recent thread on a similar problem, a rigorous mathematical justification is the implicit function theorem. You write ##y = y(x,z)##, provided ##\frac{\partial f}{\partial y} \neq 0## and then differentiate both sides of ##f(x,y,z) = 0## with respect to ##x##.
     
  6. Aug 2, 2014 #5
    Although, I suppose if you're given f(x, y, z) = 0, then the equation can be solved for y in terms of x and z in which case the partial derivative notation makes sense. Is that a reasonable explanation?
     
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