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Derivatives and fractions (relationship?)

  1. May 6, 2010 #1
    learning calculus here. got differential calculus, though it is a little foggy, and most of integral calculus, which is a little foggier. also using very unpolished precalc background, though i did give most of it a once-over. i have many questions which i can't think of, but of the top of my head...

    when learning leibniz notation it is pointed out that derivatives are different than fractions. However, some similarities, such as the chain rule, point to an obvious relationship. would someone please explain this relationship? compare/contrast? If I had to guess I would say that a derivative, being a quantification (of a function at a certain point), is something like a number, in the same way that a fraction is, and thus(?) owning an analogous internal composition. Does that mean derivatives are subject to algebraic field properties if arithmetic operators are applied? sorry just typing random words here...
     
  2. jcsd
  3. May 6, 2010 #2
    Slow down - one thing at a time.
     
  4. May 14, 2010 #3
    You speaking of the df(x)/dx ? Correct? In which case, it is an operator. You are stating take the derivative of f(x) with respect to x. After that, not quite sure. No differential operator is more like a ring I believe.
     
  5. May 15, 2010 #4

    HallsofIvy

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    Yes, it is true that [itex]dy/dx[/itex] is NOT a fraction. It is, however, the limit of a fraction so can often be treated like a fraction. For example, it is true that if y= f(x) and [itex]x= f^{-1}(y)[/itex], then
    [tex]\frac{dx}{dy}= \frac{1}{\frac{dy}{dx}}[/tex]

    You cannot prove that by just turning over the "fraction" in the denominator but you can go back before the limit in "[itex]\lim_{h\to 0} (f(x+h)- f(x)}{h}[/tex]", inverting that fraction and and taking the limit again.

    Similarly, we cannot prove the chain rule:
    [tex]\frac{df}{dx}= \frac{df}{du}\frac{du}{dx}[/tex]
    by "cancelling" the "du"s but we can go back before the limit, cancel and then take the limit again.

    That is one reason for the Leibniz notation and for then defining the "differentials" dx and dy separately- so that we can use the fact that, while a derivative is not a fraction, it can be treated like one.
     
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