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Derivative of a linear function

  1. Sep 27, 2011 #1
    Given that the derivative of a linear function is the function itself, how do I make sense of the following:
    Given f(x) = x. It's derivative is g(x) = f'(x) = 1. Is g(x) the same as f(x) in some way? Or have I got this wrong in some way. Is f'(x) really the derivative of a f(x) in the sense of the statement that "the derivative of a linear function is the function"?
     
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  3. Sep 27, 2011 #2

    HallsofIvy

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    Where were you "given that the derivative of a linear function is the function itself"? The derivative of an exponential function is the function itself, not a linear function.
     
  4. Sep 27, 2011 #3
    I have seen this confusing treatment too, rjvsngh, and this is the best sense I have been able to make of it : I think they are conflating the terms
    derivative with differential. The differential is the linear map that best approximates
    (locally) the change of a (differentiable function) , so that, e.g., a line with slope
    2x is the best local linear approximation to the change of f(x)=x^2 , so, by derivative,
    they mean differential, and the differential (at xo) is then is y-yo=2xo(x-xo). But then,
    if your function is (globally) linear to start with, then the best linear approximation, aka,
    differential , is the function itself. So, the differential of a linear function L is L itself,
    but the derivative of L itself is not L.
     
  5. Sep 28, 2011 #4
    The derivative of a linear function is the slope of the function, m. It is a constant, rather than a function. i.e. f(x) = mx + k, f'(x) = m. So in this case, g(x) isn't a function at all, but a number.
    f'(x) = g(x) is a differential equation, and therefor a whole different animal.
     
  6. Sep 28, 2011 #5

    lavinia

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    f(x) = x = 1.x

    You need to view the derivative as a mapping.
     
    Last edited: Sep 28, 2011
  7. Sep 29, 2011 #6
    But this is not the standard difference between derivative as an operator or as an
    element of the dual; it is an assignment of the differential , not the derivative.
     
  8. Sep 29, 2011 #7

    lavinia

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    Right, correct language would call it the differential.
     
  9. Sep 29, 2011 #8
    Right, Lavinia, I wish I had known when I first ran into this layout.
     
  10. Sep 30, 2011 #9
    thanks for all these explanations. following a particular reply, i did realize my question was incorrect in the usage of terms. my question originated in something i read in "Math Analysis", Apostol, 2nd ed., in the chapter on multi-variable calculus. However, looking closely, the precise statement was that "the total derivative of a linear function is the function itself" and i now realize that the total derivative as defined by Apostol and the simple derivative are different. in fact, Apostol does point this out in the text as well.

    i guess the "derivative as a number" notion arises in the serendipitous (?) fact that a linear functional on R1 amounts to multiplication by a number - the number being the so-called derivative.
     
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