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Math Help - Finding a derivative.

  1. #1
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    Finding a derivative.

    I'm not finding correctly the derivative of:

    f(x) = x^4 - 1

    I do the following:

    lim h-> 0:

    \frac{(x+h)^4 -1 -(x^4-1)}{h}

    = \frac{(x^2+h^2)^2 -x^4}{h}

    = \frac{x^4 + 2x^2h^2 + h^4 - x^4}{h}

    = \frac{2x^2h^2+h^4}{h}

    = 2x^2h+h^3

    Then, of course, when I substitute "0" for "h", I get an answer of 0.

    The correct answer is: 4x^3

    Where did I go wrong?

    Thank you!
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  2. #2
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    In the second line of your workings...

    \displaystyle (x+h)^4 = (x+h)(x+h)(x+h)(x+h) = x^4+4x^3h+\dots +h^4 \neq (x^2+h^2)^2
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  3. #3
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    Is there a simpler way to solve this while still using the definition of the derivative? I'm sure I did this before without expanding it the way you have, I just don't remember how.

    Thanks again.
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  4. #4
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    Ah, found it. I used the alternate definition of the derivative to do it.

    Works out to:

    \frac{x^4-c^4}{x-c}

    \frac{(x+c)(x-c)(x^2+c^2)}{x-c}

    (x+c)(x^2+c^2)

    Which solves my problem as well as the derivative from the standard definition.

    Thanks for the help!
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  5. #5
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    Quote Originally Posted by JennyFlowers View Post
    Is there a simpler way to solve this while still using the definition of the derivative?
    \displaystyle (x^{n})'= nx^{n-1}
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  6. #6
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    Right, but I wasn't allowed to find this derivative using the power rule. I was required to find it using either the definition or the alternate definition!
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