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Differential vs derivative explained

  1. Jul 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Really there is no problem to go with this question just something that I was discussing awhile back but was never really cleared up, what is the difference between:

    [tex] d(\int^{x}_{a} f(t)dt) = f(x) dx [/tex]

    and

    [tex] \frac{d}{dx}\int^{x}_{a} f(t) dt = f(x) [/tex]

    I understand that the top equation is called the differential form and the bottom is the derivative form and really thats about it. I was told that they were different by a factor of dx such that:

    [tex] df = \frac{d}{dx}*dx [/tex]

    but I really don't understand how that makes any difference because if dx is infinitely small how does that affect the answer of the problem?

    This arose from the book that I am using introducing The Fundamental Theorem of Calculus with the differential form and not the derivative form which evidently is the norm. I really just want a little clarification on this issue and apologize in advance if this is something quite obvious.

    Thanks!
     
  2. jcsd
  3. Jul 18, 2010 #2

    hunt_mat

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    Hard to give an answers as the definitions are quite complex. They're the basic tools of differential geometry.
     
  4. Jul 18, 2010 #3
    Ok let me rephrase then, is it at all important for me to know the difference when learning elementary calculus?
     
  5. Jul 18, 2010 #4

    hunt_mat

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    No, I never learnt about differential until I was doing a masters course. If your calculus book if telling you about differentials, get another book.
     
  6. Jul 18, 2010 #5
    ok well your reply implies that it somehow effects my answers even on basic equations in calculus to use differential definitions over derivative, is that what you meant? If not then why should I get a different book if comes out the same?
     
  7. Jul 18, 2010 #6

    hunt_mat

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    Just use the definition:
    [tex]
    \frac{df}{dx}\Bigg|_{x=a}=f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}
    [/tex]
    this is what you seem to be calling the derivative form. Everything follows from the equation above. Don't worry about differentials.
     
  8. Jul 18, 2010 #7
    ok, that is the very definition that my book illustrates but without the use of limits which are subbed for hyperreal numbers. Also I called that form the derivative form because either Dick or Mark44 called it that, is that not right?
     
  9. Jul 19, 2010 #8

    hunt_mat

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    I have never heard it called anything but the derivative.
     
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