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I have a function

  1. May 2, 2011 #1
    I have a function "f", which is a function of "T" but "T" is a function of small "t".
    Now my question is what is the derivative of "f" with respect to "t"?
  2. jcsd
  3. May 2, 2011 #2


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    Re: derivative

    What you're asking simply has no sense. Where did you encounter this?

    Basically, T could be a function [tex]T:\mathbb{R}\rightarrow \mathbb{R}[/tex] and [tex]f:\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}:T\rightarrow f(T)[/tex].

    But now there are two problems
    1) I have no clue how to define a derivative on [tex]\mathcal{C}(\mathbb{R},\mathbb{R})[/tex], I'm certain it can be done, but it's not immediately clear.
    2) f is not a function of t. The best thing you can do is to define a derivative of f w.r.t. T.

    However, you possible can do the following:
    define the function [tex]g:\mathbb{R}\times\mathcal{C}(\mathbb{R},\mathbb{R}):(t,T)\rightarrow T(t)[/tex]
    And you could possible use this to define a derivative w.r.t. t. But I'm quite sure this is not what you mean...

    Where did you encounter this, can you give me the reference??
  4. May 2, 2011 #3


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    Re: derivative

    I think saravanan13 is talking about the "chain rule":
    if y= f(T) is a function to the variable T and T itself is a function of the variable t, then we can think of y as a function of t: y= f(T(t)).

    Further, if both functions are differentiable then so is the composite function and
    [tex]\frac{dy}{dt}= \frac{df}{dT}\frac{dT}{dt}[/tex]

    So that, for example, if [itex]y= T^3[/itex] and [itex]T= 3t^2+ 1[/itex] then we can calculate that [itex]y= (3t^2+ 1)^3= 27t^6+ 27t^4+ 9t^2+ 1[itex] so that
    [tex]\frac{dy}{dt}= 182t^5+ 108t^3+ 18t[/tex]

    Or we could calculate that
    [tex]\frac{dy}{dT}= 3T^2[/tex]
    [tex]\frac{dT}{dt}= 6t[/tex]
    so that
    [tex]\frac{dy}{dt}= 3(3t^2+ 1)^2(6t)= 18t(9t^4+ 6t^2+1)= 162t^5+ 108t^2 18t[/tex]
    as before.
  5. May 2, 2011 #4
    Re: derivative

    I came across this problem in perturbation analysis formulated by Ablowitz and Kodama.
    In that T is slowly varying time and t is a fast variable.
    Thanks for your kin reply...
  6. May 2, 2011 #5
    Re: derivative

    Could you help me out how to type the mathematics formula in this forum.
    After i used some latex that give in the last icon of top left go for a preview it was not shown that i typed.
  7. May 2, 2011 #6


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    Re: derivative

    Just use the Fréchet derivative.
  8. May 3, 2011 #7
    Re: derivative

    After you click 'preview', refresh the page - it should now show you what you typed. This is a known issue on these forums.
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