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Simple Partial Differentiation Question

  1. Aug 13, 2013 #1
    1. The problem statement, all variables and given/known data

    I found this solved example in an old textbook. I don't think that the solution provided is correct. I'll be very grateful if someone could verify it.

    Question:
    xxyyzz = c

    What is [itex]\frac{∂z}{∂x}[/itex]?

    Solution Provided:
    Taking logarithms on both sides:
    zlog(z) = log(c) - xlog(x) - ylog(y)

    Differentiating w.r.t. x
    [itex](z.\frac{1}{z}+ log(z))\frac{∂z}{∂x} = -(x.\frac{1}{x} + 1.log(x))[/itex]

    2. Relevant equations


    3. The attempt at a solution

    Isn't this solution missing a [itex]\frac{∂y}{∂x}[/itex] term as:
    [itex](z.\frac{1}{z}+ log(z))\frac{∂z}{∂x} = -(x.\frac{1}{x} + 1.log(x)) - (y.\frac{1}{y} + 1.log(y))\frac{∂y}{∂x}[/itex]

    Thanks.
     
  2. jcsd
  3. Aug 13, 2013 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi Vagrant! :smile:


    ah, ∂z/∂x is defined as the derivative of z wrt x, keeping all other variables constant :wink:

    [this is so even if y is also a function of x …

    in that case, if you want the derivative of z wrt x to include the variation in y, you write it dz/dx not ∂z/∂x]​

    see eg http://en.wikipedia.org/wiki/Partial_derivative
    " all the other variables are treated as constant when taking the partial derivative …"​
     
  4. Aug 13, 2013 #3
    Ahh...ok. I get it. Thanks a lot :)
     
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