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Partial Derivative of x^y?

  1. Mar 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the first partial derivatives of:

    1. f(x,y) = x^y
    2. u = x^(y/z)


    2. Relevant equations



    3. The attempt at a solution

    f_x = y*x^(y-1)
    f_y = lnx?

    u_x = (y/z)*x^((y/z)-1)
    u_y = lnx/z?
    u_z = ylnx/z?

    I'm not really sure how to do these right. =/ I would really appreciate any help.
     
  2. jcsd
  3. Mar 1, 2009 #2

    jambaugh

    User Avatar
    Science Advisor
    Gold Member

    Your f_x is right. Your f_y is not. Look at x as a constant in this one and look up the derivative of an exponential of arbitrary base formula.

    Your u_x is right.
    Your u_y again should be treated as an exponential function base x.
    Your u_z should as well with an additional application of the chain rule.
     
  4. Mar 1, 2009 #3
    Thank you!
     
  5. Mar 1, 2009 #4
    Don't forget to use the chain-rule.

    For the y derivative of x^y:

    Let x = k, a constant.

    [tex]f(y) = k^y[/tex]

    Natural log of both sides gives:

    [tex]ln(f(y)) = ln(k^y)[/tex]

    [tex]ln(f(y)) = yln(k)[/tex]

    Differentiating...

    [tex]f'(y)/f(y) = ln(k)[/tex]

    [tex]f'(y) = f(y)ln(k)[/tex]

    Since [tex]f(y) = k^y[/tex], you now have:

    [tex]f'(y) = ln(k)k^y[/tex]

    Substituting for x...

    [tex]f_y = ln(x)x^y[/tex]
     
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