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Homework Help: Differentiation with respect to a function

  1. May 8, 2015 #1
    Hi. A very long problem brought me to a derivative in the form


    I'm assuming that


    So, is it correct to say that


    Thank you
  2. jcsd
  3. May 8, 2015 #2
    (P.S. I don't think "derivating" is a word..............)
  4. May 8, 2015 #3
    differentiation, perhaps?

    Thank you very much.
  5. May 8, 2015 #4
    Your welcome.
    It is interesting to note that this does not work with higher derivatives........
    [EDIT:- so ##d^2g(x)\neq \Big(\frac{d^2g(x)}{dx^2}\Big) dx^2##]
    Last edited: May 8, 2015
  6. May 8, 2015 #5

    Ray Vickson

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    We have
    [tex] \frac{d\, f(x)}{d\, g(x)} = \lim_{\Delta g(x) \to 0} \frac{ \Delta f(x)}{\Delta g(x)}, [/tex]
    [tex] \Delta f(x) = f(x + \Delta x) - f(x) \doteq f'(x) \Delta x, \\
    \Delta g(x) = g(x + \Delta x) - g(x) \doteq g'(x) \Delta x, [/tex]
    [tex] \frac{d f(x)} {d g(x)} = \frac{f'(x)}{g'(x)} [/tex]
    This is equivalent to what you wrote.
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