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Definition of derivative - infinitesimal approach, help :)

  1. Jun 24, 2015 #1
    Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct.

    I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the tangent to the graph which is the derivative of f at that point and tells you the slope of the function at that point. But I want the correct mathematical understanding of it.

    The book argues that

    Δy = the change in y along a curve between 2 points (I assume it's a secant)
    dy = change in y along the tangent line to that curve between 2 points (The differential)

    Then it defines that

    Δy = f(x+Δx)-f(x)
    dy = f'(x)Δx

    and mentiones that
    "let y=f(x). Suppose f'(x) exists at a cetain point x, and Δx is the infinitesimal, then Δy is infinitesimal and"

    Δy = f'(x)Δx+ εΔx

    and prooves that

    Δy/Δx ≈ f'(x)
    Δy/Δx = f'(x) + ε
    Δy = f'(x)Δx + εΔx

    My first question is This:

    Is f(x+Δx)-f(x) and f'(x)Δx+ εΔx Equal? If they both are equal to Δy then i assume they are?

    Is this the correct understanding:
    This expression Δy/Δx only approaches f'(x): Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx)/((x+εΔx-x)) ≈ f'(x)
    While this expression is equal dy/dx = f'(x)

    The difference between
    dy/dx = f'(x)
    Δy/Δx ≈ f'(x)

    is that Δy= dy+ εΔx contains that extra "εΔx" term and therefore is bigger than dy and is also the secant between 2 points along the graph, whereas dy is an infinitesimal small movement between 2 points on the tangent to the graph. Because dy=f'(x)*Δx equals the term f'(x)*Δx this tells us that dy is an infinitesimally small change in y between the point at the tangent and another point on the tangent infinitesimally close to that point. The Δx in f'(x)Δx shows us that it's a "change in y" corresponding to a infinitesimally small change in x so (x2-x1), hence Δx?

    If the f'(x) exists then the differential dy and the increment Δy MUST be infitesimal and so close together that they cannot be seen under the infinitesimal microscope.

    So my last question: I guess it's a no-go to treat the symbol dy/dx as a quotient? The book mentions that if dy=f'(x)dx and if dx ≠ 0 then we can rewrite the equation dy/dx = f'(x), is this just lucky that this "trick" works, or is it true that you can treat it like a quotient?
    Last edited: Jun 24, 2015
  2. jcsd
  3. Jun 24, 2015 #2
    Is f(x+Δx)-f(x) and f'(x)Δx+ εΔx Equal ...yes

    Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx)/((x+εΔx-x)) ≈ f'(x)
    where did you get the denominator x+εΔx-x from? I think it should just be Δx
    then the expression reads Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx)/(Δx) = f'(x)+ε
    and the ε goes to zero as Δx goes to zero.

    I guess it's a no-go to treat the symbol dy/dx as a quotient?
    yes i think it's technically incorrect to consider dy/dx as a quotient. Rather it is the limit of Δy/Δx. If the book says "that if dy=f'(x)dx and if dx ≠ 0 then we can rewrite the equation dy/dx = f'(x)", they are being technically lazy/incorrect. On the other hand casual users of calculus often treat the dy/dx as a quotient because it saves time and we don't want to write the full definition every time we manipulate the equations. But be warned this is just shorthand.
  4. Jun 24, 2015 #3


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    I have to say, if one wanted to learn infinitesimals to avoid limits, this is certainly bad news. The other bit of bad news is that the secant, when ##\Delta{x}## goes infinitesimal, becomes the tangent. This is the same as being in the limit. If all one is doing is switching the word "infinitesimal" for the word "limit", but still one must use limits or think in terms of being in the limit, I don't see what one has gained.

    I haven't read the book in detail, I did look at it about 6 years ago, but I'd be very surprised if ##dx## is allowed to be zero. That seems odd to me. If ##dx## is allowed to be zero, I would shelve this theory and just learn the usual calculus because then it is a load of tosh.
  5. Jun 24, 2015 #4
    You can treat dy/dx as a quotient. Whether this is technical precise mathematically, I don't know. But for all engineering purposes, you can treat dy/dx as a quotient. In differential equations, you habitually multiply both sides of an equation by dx.
  6. Jun 24, 2015 #5
    Thank you guys. So when is it okay to treat dy/dx as a quotient? Are there some actual rules or guilde lines? The concept of what the limit is, and what rules apply to dy/dx, and what it TRULY means has always confused me, can someone enlighten me? :)

    Just out of curiosity: What is the Full definition?
  7. Jun 24, 2015 #6
    When delta x is not equal to zero. Ussually used for problems of physical importance. For ex. Geometric Shapes (distances is one), physics problems are 2 that come to mind. Leibnezz notation ussually treats the differential as 2 separate things. A quick Google search will give you more information.
  8. Jun 24, 2015 #7


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    Christian, you have bought so many good calculus books, I suggest you read them instead of this one, for example I think you bought the 3rd edition of Thomas that MidgetDwarf recommended, and possibly you also bought Simmons. I would use them instead of this one because these infinitesimals are only going to make it more confusing.

    I'll put down here what I don't like about this book.

    1. It calls ##\Delta{x}## an infinitesimal. Perhaps it means that ##\Delta{x}## can be infinitesimal, but for me this is unfortunate and ##dx## should be the infinitesimal. ##\Delta{x}## should be a finite quantity, to have it be both is too confusing. Then it makes sense to say ##\Delta{x} \to dx## (as in, in the limit it gets there).

    2. It uses ##\epsilon## as a finite number. But I believe infinitesimal theories usually treat ##\epsilon## as an infinitesimal. It's just too confusing to read a formula where ##\Delta{x}## is infinitesimal but ##\epsilon## is finite.

    3. Infinitesimals don't make calculus easier to understand. The usual textbooks are going to have better quality because there are more of them and they are the standard.

    Sorry for that, I'll let the thread continue but I wanted to let it be known that IMHO this is not a good way to learn calculus.
    Last edited: Jun 24, 2015
  9. Jun 24, 2015 #8


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    No, the book is not sloppy, incorrect or full of tosh. It means exactly what is says and it is correct. In the context of the book, ##dy/dx## IS the quotient of ##dy## and ##dx##. But notice the difference in definitions, usually ##dy/dx## is defined as the limit of a fraction, this is not the approach here. This is one of the great accomplishments of infinitesimals.
  10. Jun 24, 2015 #9


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    Strictly mathematical terms, dy/dx is not a quotient. It is the limit of Δy/Δx as Δx -> 0. It can be safely handled as a quotient as long as you are aware of the limitations.
  11. Jun 24, 2015 #10


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    Even in standard calculus, this is wrong. The definitions of ##\Delta x## and ##dx## in standard books is exactly the same as the definitions here.

    Saying that there are more books that deal with standard calculus is not really an argument for saying those books are better. Keisler is an excellent book and an excellent way to learn calculus.
  12. Jun 24, 2015 #11
    No the last part that contains epsilon (delta x). Is considered extremely small so it is an approximation to the curve. As you take the limit In this case, there is no argument that both epsilon and delta x cannot be zero in the last term. As you take the limit of delta x as it approaches infinity. The value is extremely small so it is neglible. So you can ignore it so to speak.
  13. Jun 25, 2015 #12
    Thank you guys!! Now I think i finally understand it 100% Clearly!! - the infinitesimal approach to Calculus, so I'll try to explain it:

    This equation
    Δy =f'(x)*Δx +εΔx

    means: an infinitesimal change in y ( an infinitly small distance in y-direction between 2 points ) of a function f can be found as a corresponding change in y of the slope of a tangent to the graph, hence the f'(x)*Δx, PLUS the the infinitemimal small distance between then graph and the tangent, hence then εΔx term.

    So this Δy is the vertical y value between 2 infinitely close points on a graph of f.
    and this f'(x)*Δx is the y-value between the same 2 points on the tangent to a point on f.
    This εΔx tells us that the vertical y-value/distance between 2 points on the graph of f is + εΔx bigger than the vertical distance betwen the 2 points on the tangent to f which is f'(x)Δx

    I added two figures to show that εΔx can either be the vertical displacement between the tangent and the graph of f , or the distance between the point (x+Δx, y+Δy) on the tangent and on the grap of f. However in reality εΔx would be MUCH smaller than f'(x)*Δx, but in the figure they look almost the same in size.

    Fig 1 where εΔx is the vertical distance between the end point (x+Δx, y+Δy) of the graph of f and the tangent

    Fig 2
    where εΔx is the vertical distance between the tangent and the graph of f

    So if we look at the same equation, and we divide it by Δx,
    (Δy =f'(x)*Δx +εΔx)/Δx

    we get

    Δy/Δx =f'(x) +ε

    This equation shows us that the change in Y with respect to a change in X over a infinitesimal distance (between 2 points) only comes near to f'(x), but the actual slope at ONE point f'(x) and the slope between 2 infinitesimal pointsΔy/Δx still separated by a small ε
    Δy/Δx ≈ f'(x)

    So for the equal signs to hold we need to add ε
    Δy/Δx =f'(x) +ε

    If we take the standard parts of this ratio, we get

    st( Δy/Δx) = f'(x) =dy/dx=(f'(x)*dx)/dx = st(Δy =f'(x)*Δx +εΔx)/Δx

    standard parts
    means that if Δy/Δx is a hyperreal number ( hyperreal numbers includes real numbers + infinitesimals) - consisting of the sum of a real number f'(x) and an infinitsimal ε - then the difference between (f'(x) + ε) -(f'(x)) is infinitesimal, and so the standard part of f'(x) +ε, denoted st(f'(x)+ε) or st(Δy/Δx) is a real number f'(x) which is infinitesimally close to f'(x) +ε .

    Δy/Δx = f'(x) +ε
    st(Δy/Δx) =st( f'(x) +ε) = f'(x) = dy/dx.

    This way of writing it should also be true
    Δy/Δx =dy/dx +ε =f'(x) + ε

    Regarding the Notation Δy/Δx
    If Δy/Δx Is treated as the limit then it's a symbol and not a quotient (Thank you micromass)
    If Δy/Δx is treated as a quotient as in the book (which works) then the differential is dy= f'(x)Δx = f'(x)*dx (in the book Δx and dx are the same and both are infinitesimal)

    End Question: The book successfully treats Δy/Δx as a quotient: Does this work in ANY situation?
    Conclusion: If you want to derive a function that shows the slope of the graph of that function with some geometrical intuition, Then you find the slope betrween 2 infinitely close points Δy/Δx , and then by taking the standard parts (by removing everything which is infinitesimal) you have the derivative f'(x) of that function.So i guess that has been the fundamental problem in calculus back in the days: How do you get the instantaneous velocity/slope at one point on a function when you need 2 points to get a slope, well you just invent the infinitesimal numbers, or take the limit and then remove the delta X'es. Smart people :D
    Last edited: Jun 25, 2015
  14. Jun 25, 2015 #13


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    I don't understand what you're doing in Figure 2. The line you show is not a tangent to the curve. A line that is tangent to a curve at some point touches the curve at that point.
  15. Jun 25, 2015 #14


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    I think you mean ##dy/dx## and not ##\Delta y/\Delta x##. Even in standard situations, ##\Delta y/\Delta x## is a number. It is ##dy/dx## that is usually defined as a limit, but that here is succesfully defined as a quotient.

    Assuming you mean ##dy/dx##, yes it always works.

    Yes, indeed, that was a very smart solution by some of the smartest people ever to live. But notice that the approach had problems back then. The problem was that nobody really knew what infinitesimals were and what their exact properties are. It is only recently that those questions were answered. Their properties are well-known now and follow mostly from the transfer principle (which is one of the most awesome results out there). This is why the standard approach has been invented: to eliminate any issue with infinitesimals which were seen as badly behaved. So the difference between the standard and the nonstandard approach is that the standard approach is able to formulate everything in function of real numbers, and they do not need extra infinitesimal or infinite numbers.
  16. Jun 25, 2015 #15
    Ahh yes, i see your point: In the book is written "When f'(x) exists then The curve y=f(x) and the tangent line at (x,y) are so close to each other that they cannot be distinguished under an infinitesimal microscope." So i assumed that εΔx could also be interpreted as the distance between the tangent line and the function, but i see now that it makes no sense, so εΔx is to be interprete as figure one. Thank you for clarifying :)
  17. Jun 25, 2015 #16
    Yes thank you for pointing that out. dy/dx is what i meant :)

    P.s I really like this approach to calculus, it's a great book you recommended back in my other post on calculus book . Thank you :)
  18. Jun 25, 2015 #17
    So just to be clear with the last part

    Δy/Δx Is the slope of a secant between 2 points of the function f
    dy/dx is the change in y with respect to x On the tangent line.
    dy =f'(x)dx is called the differential and is just a way of rewriting dy/dx = f'(x) and it works,
    UPDATE from micromass: so dy and dx are changes in the tangent line and means. the tangent line dy is proportional to the change in the domain dx, and the proportionality factor is f'(x).

    just a quick last question:
    If dy/dx means a change in y with respect to x, at a specific point, then what exactly does dy=f'(x)*Δx mean? Is it just a notation, or does it have a meaning different from dy/dx?
    Last edited: Jun 25, 2015
  19. Jun 25, 2015 #18


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    Do you mean ##dy = f^\prime (x) dx##? The ##dy## and ##dx## are changes on the tangent line. It means that the change of the tangent line ##dy## is proportional to the change in the domain ##dx##. In this sense, ##dy## and ##dx## are numbers that are related as ##dy/dx = f^\prime(x)## or equivalently as ##dy= f^\prime(x) dx##.
  20. Jun 25, 2015 #19


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    Also, and more importantly in regard to your figure 1, dy/dx is the slope of the tangent line at the point (x, f(x)).
  21. Jun 25, 2015 #20


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    To back up what I was saying about the possibility of dx being zero, I quote from here:

    This is obviously historical but one can see that originally dx was not allowed to be zero. The points got super duper close but didn't ever land on each other, they were still separated but were infinitely close. And they were close enough that their secant *was* the tangent. That said, this point of view became obsolete when calculus was made rigorous and the limit concept became the standard.

    And I must say, I really like the limit concept and would definitely rely on it. I think it is the right notion and one shouldn't need an alternative because limits work perfectly well and are, I believe, easier to understand than infinitesimals.
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