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Notation for derivatives

  1. Mar 30, 2008 #1
    1. The problem statement, all variables and given/known data

    I cannot seem to find my textbook and we just started derivatives. Can anyone tell me when I would use the notation dy/dx as opposed to something like f'(x)??

    Thanks!
     
  2. jcsd
  3. Mar 30, 2008 #2
    f'(x) is just function notation. If you're given an f(x), it's equally correct to use f'(x) or d(f(x))/dx, for example. In the case of f'(x), the derivative with respect to x is implied.
     
  4. Mar 30, 2008 #3
    So it wouldn't ever matter which one I use? I could use both?
     
  5. Mar 31, 2008 #4
    You should stick with one for consistency. There's nothing wrong with using y' over dy/dx, or f '(x) over d(f(x))/dx; Just be consistent.
     
  6. Mar 31, 2008 #5

    rock.freak667

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    Not loosely in a sense.

    For example, if you have

    [itex]y=x^2[/itex]
    You would write [itex]\frac{dy}{dx}[/itex] or [itex]y'[/itex] = [itex]2x[/itex] and NOT [itex]f'(x)=2x[/itex]
     
  7. Mar 31, 2008 #6

    nicksauce

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    The dy/dx notation is nice when you're actually manipulating these as if they're actually numbers. It's a physicist's favorite trick (but a mathematician's worst nightmare). I for one am not a fan of f'(x), except when there are higher order derivatives involved.
     
  8. Mar 31, 2008 #7
    I'd get used to both forms, as they seem to turn up alternately, at least in my experience.

    For example the quotient rule:

    Leibniz notation:

    [tex]\frac{d}{dx}\left (\frac{u}{v}\right ) = \frac{\frac{du}{dx}\cdot v-u\cdot\frac{dv}{dx}}{v^2}[/tex]

    Newtonian notation:

    [tex]h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}[/tex]

    or in shorthand notation:

    [tex]h'=\frac {f'\cdot g-f\cdot g'}{g^2}[/tex]

    You might see something similar as well which is a Newtonian notation:

    [tex]\dot{h}(x)=\dot{f}(x)+\dot{g}(x) \longrightarrow \ddot{h}(x)=\ddot{f}(x)+\ddot{g}(x)[/tex]

    or

    [tex]h\,\dot{}\,(x)=f\,\dot{}\,(x)+g\,\dot{}\,(x) \longrightarrow h\,\ddot{}\,(x) = f\,\ddot{}\,(x) + g\,\ddot{}\,(x)[/tex]

    occasionally but these aren't used very often.

    Which is the same as saying:

    [tex]\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}\longrightarrow \frac{d^2}{dx^2}(u + v)=\frac{d^2u}{dx^2}+\frac{d^2v}{dx^2}[/tex]

    in Leibniz notation.

    They look more complicated between forms sometimes, but once you get the idea they're pretty obvious.
     
    Last edited: Mar 31, 2008
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