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Simple question about derivatives

  1. Mar 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose you have a function [tex]f(x)[/tex].
    You take the derivative with respect to x and evaluate it at some point [tex]x_0[/tex].
    i.e. [tex]{{df(x_{0})}\over{dx}}[/tex] (e.g. first coeff in taylor series)
    Is this the same as changing the argument of f to x_0, i.e. writing [tex]f(x_0)[/tex],
    then just taking the derivative with respect to x_0?
    In fewer words, can I write

    2. Relevant equations

    3. The attempt at a solution
    If I just consider a bunch of examples, it always seems to be true. It seems like a trivial fact, but I would just like someone to confirm this for me.

  2. jcsd
  3. Mar 3, 2008 #2

    [tex]\frac{df(x_0)}{dx}[/tex] may be read as the derivative of f(x) with respect to x, evaluated at [tex]x = x_0[/tex]. I am confused as to how you were taking the derivative of something with respect to a constant.
  4. Mar 3, 2008 #3
    You're right. It seems highly unorthodox.
    Ok. Consider the taylor series (keeping only first order)
    [tex]q(\Delta t)=q(0)+{{\partial H(q,p)}\over{\partial p}}\Delta t[/tex]
    [tex]p(\Delta t)=p(0)-{{\partial H(q,p)}\over{\partial q}}\Delta t[/tex]
    where the derivatives are evaluated at t=0. I used hamilton's equation.
    [tex]{{dq}\over{dt}}={{\partial H}\over{\partial p}}[/tex]
    [tex]{{dp}\over{dt}}=-{{\partial H}\over{\partial q}}[/tex]
    Now regard this as a transformation from the old coords q(0), p(0) to the new coords q(Delta t), p(Delta t).
    What is the Jacobian for this tranformation?
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