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Math Help - Differentiation from first principles

  1. #1
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    Differentiation from first principles

    I'm stick on differentiating, from first principles, the following function:

    f(x) = (3x-1)/(x+2)

    I plug that into:

    [f(a+h) - f(a)]/h

    and I get:

    7h/(x+h+2)(x+2)

    But when I use the quotient rule to do it, I end up with:

    7/(x+2)^2


    I've been trying to get rid of that bothersome h, but I just haven't found a way to do it! I have a feeling that I'm overlooking something really simple. Any help appreciated.
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  2. #2
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    Well, you have to divide by h before you take the limit as h goes to zero, right?
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  3. #3
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    It's probably easiest to simplify the fraction before trying to differentiate it.

    f(x) = \frac{3x - 1}{x + 2}

     = \frac{3x + 6 - 7}{x + 2}

     = \frac{3(x + 2)}{x + 2} - \frac{7}{x + 2}

     = 3 - \frac{7}{x + 2}.


    Therefore f(x + h) = 3 - \frac{7}{x + h + 2}.


    f(x + h) - f(x) = \left(3 - \frac{7}{x + h + 2}\right) - \left(3 - \frac{7}{x + 2}\right)

     = \frac{7}{x + 2} - \frac{7}{x + h + 2}

     = \frac{7(x + h + 2) - 7(x + 2)}{(x + 2)(x + h + 2)}

     = \frac{7x + 7h + 14 - 7x - 14}{(x + 2)(x + h + 2)}

     = \frac{7h}{(x + 2)(x + h + 2)}.


    So \frac{f(x + h) - f(x)}{h} = \frac{\frac{7h}{(x + 2)(x + h + 2)}}{h}

     = \frac{7h}{h(x + 2)(x + h + 2)}

     = \frac{7}{(x + 2)(x + h + 2)}.


    So \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} = \lim_{h \to 0}\frac{7}{(x + 2)(x + h + 2)}

     = \frac{7}{(x + 2)(x + 2)}

     = \frac{7}{(x + 2)^2}.
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  4. #4
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    Quote Originally Posted by blackdragon190 View Post
    I'm stick on differentiating, from first principles, the following function:

    f(x) = (3x-1)/(x+2)

    I plug that into:

    [f(a+h) - f(a)]/h

    and I get:

    7h/(x+h+2)(x+2)
    No. That "h" in the numerator should not be there.

    But when I use the quotient rule to do it, I end up with:

    7/(x+2)^2


    I've been trying to get rid of that bothersome h, but I just haven't found a way to do it! I have a feeling that I'm overlooking something really simple. Any help appreciated.
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  5. #5
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    Silly me! I got so caught up in playing around with the fraction that I forgot that I still had to divide by h. I just knew that it was something silly.

    Thanks for the help
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  6. #6
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    You're welcome for whatever I contributed. Have a good one!
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  7. #7
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    Seems like I forgot about the limit too...doh..
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