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Chain rule notation - can Leibniz form be made explicit?

  1. Dec 21, 2009 #1
    Hi there, I'm a new user to the forums (and Calculus) and I 'm hoping you can give me your opinion on my chain rule form below. When learning the chain rule, I was taught two forms. This form:
    As well as the Leibniz form
    [tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}[/tex] where [tex]y=f(u)[/tex] and [tex]u=g(x)[/tex]

    I prefer the Leibniz notation, except that it requires you to understand that [tex]y=f(u)[/tex] and [tex]u=g(x)[/tex], to really understand the d/dx expression.

    So my question is if there is a way to make Liebniz more explicit? ie. Does the following make sense? Is it correct?
    [tex]\frac{d}{dx}f(g(x)) = \frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}[/tex]
  2. jcsd
  3. Dec 21, 2009 #2


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    Welcome to PF!

    Hi seand! Welcome to PF! :smile:
    Yes, that's fine.

    And I agree, the Leibniz form is easier to use, simply because you can "cancel" in it, just like ordinary fractions. :wink:
  4. Dec 21, 2009 #3


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    dg/dx = g'(x). The notations are interchangable. Just get used to both of them.
  5. Dec 22, 2009 #4


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    It's possible, but a bit confusing to me. I prefer the non-Leibniz notation anyway. It looks more like (in fact, is) the more general chain rule [tex]\mbox{D}_x\left(g \circ f\right) = \mbox{D}_{f\left(x\right)}\left(g\right) \circ \mbox{D}_x\left(f\right)[/tex].

    I find this clearer
    [tex]\frac{d}{dx}f(g(x)) = \left.\frac{df(y)}{dy}\right|_{y=g(x)}\frac{dg(x)}{dx}[/tex]
    but the nice thing about the expression [tex]\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}[/tex] is that it behaves like fractions, which you lose this way.
  6. Dec 22, 2009 #5


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    Leibniz notation really prefers that, notationally, you are manipulating algebraically-dependent variables rather than functions.
  7. Dec 22, 2009 #6
    Thank you Hurkyl, Landau, mathman and Tiny Tim for the replies and especially the suggestions. It's interesting to know that I could express the chain rule the way I did, but that there are other, more normal ways to do it.
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