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Differentiation of a function with respect to itself

  1. Aug 20, 2009 #1
    In one of my electronics textbooks I have the following equation related to feedback in amplifiers:

    [tex]K_f = \frac{K}{1-K\beta}[/tex]

    [tex]\frac{dK_f}{K_f} = \frac{1}{1-K\beta}\frac{dK}{K}[/tex]

    I'm not sure how this was derived - how was Kf differentiated with respect to itself?
     
  2. jcsd
  3. Aug 20, 2009 #2

    jgens

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    Gold Member

    [itex]K_f[/itex] wasn't differentiated with respect to itself, it was differentiated with repect to [itex]K[/itex]. Here's what they did . . .

    [tex]K_f = \frac{K}{1 - \beta K}[/tex]

    [tex]\frac{\mathrm{d}K_f}{\mathrm{d}K} = \frac{(1 - \beta K) + (\beta K)}{(1 - \beta K)^2}[/tex]

    [tex]\frac{\mathrm{d}K_f}{\mathrm{d}K} = \frac{1}{(1 - \beta K)^2}[/tex]

    [tex]\frac{\mathrm{d}K_f}{K_f} = \frac{\mathrm{d}K}{K(1 - \beta K)}[/tex]

    Basically, it's just an application of the quotient rule for differentiation.
     
  4. Aug 20, 2009 #3
    Ah, I see now. They took the derivative of Kf with respect to K, and then expressed that derivative as a ratio to get dKf/Kf. Thank you!
     
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