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    (Original post by StarvingAutist)
    Oh, um, wow :mmm: that was sure convincing..
    Maybe I'm just deluded then :lol:

    But why don't they include it? The space taken up by that table & paragraph could easily have been used :eek:
    How would you, say, prove that \dfrac{d}{dx} x^{0.3} = 0.3 x^{-0.7}?
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    (Original post by DFranklin)
    How would you, say, prove that \dfrac{d}{dx} x^{0.3} = 0.3 x^{-0.7}?
    Prove the case for xn​ and shove in 0.3

    edit: in case you wanted a better answer (though with shoddy notation)
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    (Original post by StarvingAutist)
    Oh, um, wow :mmm: that was sure convincing..
    Maybe I'm just deluded then :lol:

    But why don't they include it? The space taken up by that table & paragraph could easily have been used :eek:
    They hadn't introduced the binomial theorem at that point, so they couldn't give a proof of it.
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    (Original post by StarvingAutist)
    Prove the case for xn​ and shove in 0.3

    edit: in case you wanted a better answer (though with shoddy notation)
    You've used the binomial theorem (in the case of a non-integral power). The standard proof of this requires knowledge of the derivative of x^0.3, so your argument is circular.
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    (Original post by DFranklin)
    You've used the binomial theorem (in the case of a non-integral power). The standard proof of this requires knowledge of the derivative of x^0.3, so your argument is circular.
    Damn it, I forgot that. How retarded. In that case, I don't know.
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    (Original post by StarvingAutist)
    Damn it, I forgot that. How retarded. In that case, I don't know.
    It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).
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    (Original post by StarvingAutist)
    Damn it, I forgot that. How retarded. In that case, I don't know.

    (Original post by DFranklin)
    It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).
    More fundamentally, it's no good talking about the derivative of a function until you can actually say how the function itself works.

    How do you know what x^{0.3} means for a real number x? OK, this isn't too bad because 0.3 is rational, so we can argue (correctly) that it's the 10th root of x cubed. But what do we do when confronted by x^{\sqrt{2}} or x^r for general real r?

    There are good reasons why Bostock and Chandler (and comparable texts) say "take this [result] on trust" when introducing rules for things like differentiation
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    (Original post by DFranklin)
    It's not at all trivial (it's basically all 1st year university material, but even if you know what you're doing, doing it all properly is probably a couple of hours work).
    How do you do it? Use xn = eln(x^n) = enlnx and prove the chain rule? How much more is there?
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    (Original post by davros)
    More fundamentally, it's no good talking about the derivative of a function until you can actually say how the function itself works.

    How do you know what x^{0.3} means for a real number x? OK, this isn't too bad because 0.3 is rational, so we can argue (correctly) that it's the 10th root of x cubed. But what do we do when confronted by x^{\sqrt{2}} or x^r for general real r?

    There are good reasons why Bostock and Chandler (and comparable texts) say "take this [result] on trust" when introducing rules for things like differentiation
    Yeah, very true... I suppose it's a good thing I'm going for physics; it seems I'd fail a maths degree! It's all interesting though, and things just get deeper the harder you look
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    (Original post by StarvingAutist)
    How do you do it? Use xn = eln(x^n) = enlnx and prove the chain rule? How much more is there?
    Also need to define e^x and ln x, show that the definitions do what you expect and show that defining x^n as you describe does what you expect.

    Edit: The last bit includes what davros was alluding to when talking about x^pi etc.
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    (Original post by StarvingAutist)
    How do you do it? Use xn = eln(x^n) = enlnx and prove the chain rule? How much more is there?
    What are you asking about now?

    If you're wanting to differentiate x^r you can define the exponential function (and prove that it has an inverse, ln), and then define x^r = exp(rlnx), using the differentiability of the exponential function (which you also need to prove!).

    If you're talking about proving the binomial expansion for general exponent, again you have to define what a power series is, define radius of convergence, prove that differentiability makes sense within the radius of convergence and THEN use the limit definition of the derivative to show that the power series expansion gives you the function you started with!
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    (Original post by DFranklin)
    Also need to define e^x and ln x, show that the definitions do what you expect and show that defining x^n as you describe does what you expect.

    Edit: The last bit includes what davros was alluding to when talking about x^pi etc.
    (Original post by davros)
    What are you asking about now?

    If you're wanting to differentiate x^r you can define the exponential function (and prove that it has an inverse, ln), and then define x^r = exp(rlnx), using the differentiability of the exponential function (which you also need to prove!).

    If you're talking about proving the binomial expansion for general exponent, again you have to define what a power series is, define radius of convergence, prove that differentiability makes sense within the radius of convergence and THEN use the limit definition of the derivative to show that the power series expansion gives you the function you started with!
    Oh jesus, what a can of worms I opened! I can see why no-one bothers to prove it now :lol: I suppose doing what I did is just as unsatisfactory as saying 'trust me, this works'
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    Once you've decided,absolutely you will. Mathematics is an anyday carear with lot of trials. Just be focused.
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    (Original post by AkashdeepDeb)
    Follow these 3 steps, they really do work!

    1) Never miss a single lecture, always attend one even if the topic taught is easy.
    2) Before attending a class, always get to know what is being taught, ie. learn a sizable chunk of material prior to the lecture, so that you know what you are going to learn; this boosts up confidence.
    3) Get more books and just practice. Practice, practice and practice.

    Good luck!
    Ideally yes but certainly (2) requires extreme dedication - I never did this and was fine - don't overwork - this is extremely important in maths.

    for 1) in first year I attended around 75% of lectures but its been increasing every year!

    3) practice very true
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    (Original post by DFranklin)
    Also need to define e^x and ln x, show that the definitions do what you expect and show that defining x^n as you describe does what you expect.

    Edit: The last bit includes what davros was alluding to when talking about x^pi etc.
    How would one go about this?
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    (Original post by MathMeister)
    How would one go about this?
    There're a lot of possibilities. http://en.wikipedia.org/wiki/Charact...ntial_function
 
 
 
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