1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Taking Partial Derivatives

  1. Sep 2, 2008 #1
    1. The problem statement, all variables and given/known data
    If [tex] z = ax^2 + bxy + cy^2 [/tex] and [tex] u = xy [/tex], find [tex] \left(\frac{\partial z}{\partial x}\right)_{y} [/tex] and [tex] \left(\frac{\partial z}{\partial x}\right)_{u} [/tex] .

    2. Relevant equations
    I have Euler's chain rule, the "splitter" and the "inverter" for dealing with partial derivatives.

    3. The attempt at a solution
    I think finding [tex] \left(\frac{\partial z}{\partial x}\right)_{y} [/tex] is easy.
    [tex] \left(\frac{\partial z}{\partial x}\right)_{y} = 2ax + by [/tex]

    However, I do not know how to begin to find [tex] \left(\frac{\partial z}{\partial x}\right)_{u} [/tex] because of the extra function u. One thought is substituting u for xy in the second term on the right side of the original equation ( i wouldn't know how to differentiate it though).

    Any kind of direction would be helpful.

    Thanks in advance.
  2. jcsd
  3. Sep 2, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Replace y in the function by u/x.
  4. Sep 2, 2008 #3
    All right, let me try your suggestion:

    [tex] z = ax^2 + bx\left(\frac{u}{x}\right) + c\left(\frac{u}{x}\right)^2 [/tex]
    [tex] z = ax^2 + bu + \frac{cu^2}{x^2} [/tex]

    [tex] \left(\frac{\partial z}{\partial x}\right)_{u} = 2ax - \frac{2cu^2}{x^3} [/tex]

    I guess I can resubstitute to get:
    [tex] \left(\frac{\partial z}{\partial x}\right)_{u} = 2ax - \frac{2cy^2}{x} [/tex]

    Is this correct?

    Thanks in advance.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Taking Partial Derivatives
  1. Partial derivative (Replies: 21)

  2. The partial derivative (Replies: 2)