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    When you diffrentiate 2x2 Why do we get 4x as the diffrential?

    Why does the method work? I was just curious.

    NOTE: I am not asking how we diffrentiate. I want to know why it works.
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    (Original post by SirRaza97)
    When you diffrentiate 2x2 Why do we get 4x as the diffrential?

    Why does the method work? I was just curious.

    NOTE: I am not asking how we diffrentiate. I want to know why it works.
    https://en.wikipedia.org/wiki/Derivative
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    (Original post by SirRaza97)
    When you diffrentiate 2x2 Why do we get 4x as the diffrential?

    Why does the method work? I was just curious.

    NOTE: I am not asking how we diffrentiate. I want to know why it works.
    \displaystyle \lim_{h \to 0} \frac{2(x+h)^2 - 2x^2}{h}
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    Anything but Wikipedia?
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    (Original post by SirRaza97)
    Anything but Wikipedia?
    It's explained clearly.
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    (Original post by SirRaza97)
    When you diffrentiate 2x2 Why do we get 4x as the diffrential?

    Why does the method work? I was just curious.

    NOTE: I am not asking how we diffrentiate. I want to know why it works.
    Broadly, because of the binomial expansion theorem:

     \displaystyle \frac{d}{dx}x^n = \lim_{h\rightarrow 0} \frac{(x+h)^n-x^n}{h}

    When you expand out with the binomial theorem, the only term that survives is the term  \frac{hnx^{n-1}}{h} = nx^{n-1}
    Spoiler:
    Show

    Another method of showing this is using the definition of  e^x as "the function that equals it's own derivative. That's perfectly self contained a definition, and you then then say:
     x^n = e^{nln(x)} \implies \frac{d}{dx}x^n = \frac{n}{x}e^{nln(x)} = nx^{n-1}

    However, that requires you to know something about: log laws, the chain rule, inverse functions/implicit differentiation. Some of that stuff is C3, wherease the binomial expansion theorem is taught in C2.
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    (Original post by SirRaza97)
    Anything but Wikipedia?
    https://www.khanacademy.org/math/dif...new-hd-version
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    (Original post by lerjj)
    Broadly, because of the binomial expansion theorem:

     \displaystyle \frac{d}{dx}x^n = \lim_{h\rightarrow 0} \frac{(x+h)^n-x^n}{h}

    When you expand out with the binomial theorem, the only term that survives is the term  \frac{hnx^{n-1}}{h} = nx^{n-1}
    What do you mean by survives?

    We have (x+h)^n - x^n = hnx^{n-1} + o(h^2).
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    (Original post by SirRaza97)
    When you diffrentiate 2x2 Why do we get 4x as the diffrential?

    Why does the method work? I was just curious.

    NOTE: I am not asking how we diffrentiate. I want to know why it works.
    I recommend you watch this video:

    https://www.youtube.com/watch?v=Ayf9gKwjXlY
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    (Original post by Zacken)
    What do you mean by survives?

    We have (x+h)^n - x^n = hnx^{n-1} + o(h^2).
    Sorry, by "survive" I mean that in the limit h \rightarrow 0 that's the only term that doesn't vanish (after dividing by h in every term, of course).
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    (Original post by lerjj)
    Sorry, by "survive" I mean that in the limit h \rightarrow 0 that's the only term that doesn't vanish (after dividing by h in every term, of course).
    Ah, okay - I wasn't sure because you left an \frac{h}{h} in your answer so was wondering whether you'd taken the limit or not yet. S'alright now, thanks.
 
 
 
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