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Seemingly Simple Derivative (as a limit) Problem

  1. Nov 20, 2012 #1
    I'm having trouble showing the following:

    lim [f(ax)-f(bx)]/x = f'(0)(a-b)
    x→0

    I feel like this should be really easy, but am I missing something? I tried to use the definition of the derivative, but I know I can't just say f(ax)-f(bx) = (a-b)f(x).

    Any ideas?
     
  2. jcsd
  3. Nov 20, 2012 #2

    arildno

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    Try to add zero in your numerator in the shape f(0)-f(0), and see if you can rearrange it in a clever manner.
     
  4. Nov 20, 2012 #3
    You mean so that I get:

    [lim f(ax) - f(0)]/x - [lim f(bx) - f(0)]/x
    x→0 x→0

    I had thought about that, but I still don't see how that gives me af'(0) - bf'(0)...
     
  5. Nov 20, 2012 #4

    arildno

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    Think chain rule..
     
  6. Nov 20, 2012 #5

    arildno

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    A further hint:
    Let g(x)=ax. Then, g(0)=0
     
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