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Thread: Derivatives

  1. #1 Derivatives 
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    I am having some trouble understanding derivatives.

    To begin, I want to know why they assigned the change in y to "f(c + delta x) - f(c) over the change in x, (c + delta x) - c. And how they simplified the denominator to merely delta x.

    Why would you refer to y as x? Shouldn't they change the x in the numerator to y?


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    The derivative of a function at a given point (in this case ) is its instantaneous rate of change. So on top we have which is an arbitrarily small distance away from . We subtract from this itself to obtain the tiny amount by which the function value (, or ) changes. I've never actually realized this before but now upon examining this I see what this definition of the derivative actually shows and it's really cool.

    The numerator is equal to when we change our input () by an arbitrarily small amount, . We divide by because this is the change in . So what we have is (at ), which is the rate of change of the function, which is the interpretation of the derivative.

    As for the denominator you have, , the s simply cancel each other out. You're doing the same thing here as you are to the s in the numerator, but here the and go unchanged because they're not the input for a function and the s can simply be cancelled.


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    Chemboy gave an excellent explaination, but I am a visual kind of guy so I created some pictures to help you understand what is going on.

    In all these pictures, we have a non linear function as our main function.
    In this first picture, we can find the average rate of change between c and d by finding the slope of a secant line drawn from the points (c , f(c) ) and (d , f(d) ).


    If we wanted to find the rate of change a bit closer to c, we could bring our second point d closer to c and thus get a secant line that more closely resembles the slope of the main function at c.


    Bringing d even closer to c makes the slope even more like that at c.


    So instead of nameing a second point d, lets instead refer to the second point as c plus the distance from c to d. We will now write this second point as:

    which means c plus the distance from c to d. We can also, instead of writing the y value as f(d), write


    As previously stated, as we bring d closer in to c, the slope becomes more like the exact value of the slope at c, or in other words, the smaller delta x gets, the more accurate our slope gets.

    If you look at the pictures, this formula:

    is nothing but the formula for finding a slope of a line:
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    Thanks. Your explanations even after a quick look cleared things up allot. I will analyze it more tomorrow.
    Of all the wonders in the universe, none is likely more fascinating and complicated than human nature.

    "Two things are infinite: the universe and human stupidity; and I'm not sure about the universe."

    "Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence"

    -Einstein

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    Something I'd like to add... This definition should really be written as



    to indicate that is arbitrarily small.
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    Ok.....so looking at the equation, the whole f(c + delta x) - f(c) is just a formality of sorts to completely explain an already known logical process...which is that your point declaring the beginning of the change in x is likely not at the origin, therefore meaning that there is an amount before it that you do not want implied into the equation, hence the -f(c) part. From what I can tell, you would only need (delta x) in the numerator if your change started at the origin?
    Of all the wonders in the universe, none is likely more fascinating and complicated than human nature.

    "Two things are infinite: the universe and human stupidity; and I'm not sure about the universe."

    "Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence"

    -Einstein

    http://boinc.berkeley.edu/download.php

    Use your computing strength for science!
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    Quote Originally Posted by Cold Fusion
    From what I can tell, you would only need (delta x) in the numerator if your change started at the origin?
    You would be wrong to think that.

    Consider it like this: (c + d) - c = d

    And this is true wherever c appears on the x-axis, and however small d might be.

    Delta-x simply means an arbitrarily small increase in the x-value; an increase so arbitrarily small that we are not able to assign a proper number to it, so we just call it "a little tiny increase in the x-value" which we write delta-x.

    This would be true whatever value x happened to have.
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  9. #8 Re: Derivatives 
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    Quote Originally Posted by Cold Fusion
    I am having some trouble understanding derivatives.

    To begin, I want to know why they assigned the change in y to "f(c + delta x) - f(c) over the change in x, (c + delta x) - c. And how they simplified the denominator to merely delta x.

    Why would you refer to y as x? Shouldn't they change the x in the numerator to y?
    It is a DEFINITION. You can do anything that you want to in a definition.

    The important things are the consequences of the defnition. In the cases the consequences are called differential calculcus and it has been rather successful.
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  10. #9  
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    Why is it said that the difference is "arbitrarily small"? Aren't we dealing with the difference in distance between our two points?
    Of all the wonders in the universe, none is likely more fascinating and complicated than human nature.

    "Two things are infinite: the universe and human stupidity; and I'm not sure about the universe."

    "Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence"

    -Einstein

    http://boinc.berkeley.edu/download.php

    Use your computing strength for science!
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  11. #10  
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    Quote Originally Posted by Cold Fusion
    Why is it said that the difference is "arbitrarily small"? Aren't we dealing with the difference in distance between our two points?
    Yes, we are dealing with the distance between two points.

    Arbitrarily small means small in comparison to whatever scale is being employed on the x-axis, which is itself quite arbitrary. If your x-axis is measured in inches then a small increment might be one thousandth of an inch, but if your x-axis is measured in astronomical units then a small increment might be ten thousand light years. If you then change the scale on the x-axis to be half-astronomical units the same increment as before has become twenty thousand light years. How small it is, is based on your arbitrary choice of scale.

    If this confuses the issue for you then just think of it as being small.

    It is also worth reminding yourself that delta-x and delta-y are just numbers. They are not written as decimal numbers, and they generally have values so small that assigning values to them would be pretty pointless, but they are nevertheless just numbers.
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  12. #11  
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    It always helped me to use the equation



    The pictures that Demen posted were also the way I was taught to visualize the derivative at a given point.

    If your interested in learning Calculus basics, I can point you in the direction of a superb well hidden free website with hours of video lectures used at my University.

    There is also some advanced video's covering Calculus subjects like double and triple integrals.
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  13. #12  
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    post a link
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  14. #13  
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    Quote Originally Posted by Demen Tolden
    post a link
    Your wish is granted

    http://www.math.lamar.edu/faculty/ma...lculusone.html

    and

    http://www.math.lamar.edu/faculty/ma...ulusthree.html

    It's not possible to navigate to the site other than directly typing in that address. That is why it is well hidden. I don't know why he doesn't make the site more accessible to people.

    The first couple of lectures on the Calculus one site are review. You have to scroll down to see the lecture links, all open in Windows media player and are downloadable.

    I'm in his calculus three class right now. Were just finishing up double integrals with polar coordinates. Its pretty interesting stuff.

    Note that the calculus three lectures are much more theoretical in nature than the calculus one lectures.

    Also you can go to "pauls online math notes" another faculty from my University. [/tex]
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  15. #14  
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    Quote Originally Posted by Cold Fusion View Post
    Why would you refer to y as x? Shouldn't they change the x in the numerator to y?
    It is maybe ... . But, we cannot say it as its derivation ... because the derivation of y respect to x means the gradation y respect to x ... .
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  16. #15  
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    Quote Originally Posted by trfrm View Post
    It is maybe ... . But, we cannot say it as its derivation ... because the derivation of y respect to x means the gradation y respect to x ... .
    Just as a side note - this thread is four years old (!), and as far as I am aware none of the participants are still around on this forum. It is best you limit yourself to posting on more recent threads.
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    I'm sorry ... . Thank you for your advice ... .
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