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Difference between f '(2x) and [f(2x)] ' ?

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data
    What is the difference between f '(2x) and [f(2x)] ' ?

    If we integrate, is there any difference between
    ∫ f '(2x) dx and ∫ [f(2x)] ' dx?

    2. Relevant equations
    N/A

    3. The attempt at a solution
    N/A

    Can someone please explain? I would really appreciate!:smile:
     
  2. jcsd
  3. Sep 24, 2009 #2

    Office_Shredder

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    [f(2x)]' is the derivative of the function f(2x)

    f'(2x) is the derivative of the function f(x) applied at the point 2x. An example may help:

    f(x) = sin(x). Then f'(x) = cos(x). So f'(2x) = cos(2x)

    On the other hand, f(2x)= sin(2x) So [f(2x)]' = 2cos(2x).
     
  4. Sep 24, 2009 #3
    How would you write that in Leibniz notation?
     
  5. Sep 24, 2009 #4

    Office_Shredder

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    I've often seen it written as:

    [tex]f'(2x) = \frac{df}{dx} \Big |_{2x}[/tex]

    The vertical bar meaning 'evaluated at'. So this is the derivative of f evaluated at the point 2x. For the other case

    [tex][f(2x)]' = \frac{d[f(2x)]}{dx}[/tex]
     
  6. Sep 25, 2009 #5
    So you wouldn't write
    [tex]
    \frac{d\,f}{d\;2x}
    [/tex]
    ??
     
  7. Sep 25, 2009 #6

    Avodyne

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    I don't agree. It seems to me that the prime on the left means "take the derivative with respect to the argument", and the argument is 2x; the d/dx on the right means "take the derivative with respect to x".

    Ambiguity is removed via

    [tex]f'(2x) = \frac{df(y)}{dy} \Big |_{y=2x}[/tex]
     
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