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Thread: Problem with Diff'ing a rational expression

  1. Nov 18th 2007, 07:17 AM #1
    BClary
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    Problem with Diff'ing a rational expression

    Keep getting all twisted up in the algebra in finding the first and second derivative of the following

    x(third)-4x(squared)/(x-2)(squared)

    F’(x)
    (x(third)-4x(squared))’(x-2)(squared)-(x(third)-4x(squared))((x-2)(squared))’/(x-2)(fourth)

    than

    (3x(squared)-8x)(x-2)(squared)-(x(third)-4x(squared)(2)(x-2)(1)/(x-2)(fourth)

    than

    (x-2)[(3x(squared)-8x)(x-2)-2x(third)+8x(squared)]/(x-2)(fourth)

    now stuck

    any help on further factoring would be appreciated
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  2. Nov 18th 2007, 07:45 AM #2
    topsquark
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    Quote Originally Posted by BClary View Post
    Keep getting all twisted up in the algebra in finding the first and second derivative of the following

    x(third)-4x(squared)/(x-2)(squared)

    F’(x)
    (x(third)-4x(squared))’(x-2)(squared)-(x(third)-4x(squared))((x-2)(squared))’/(x-2)(fourth)

    than

    (3x(squared)-8x)(x-2)(squared)-(x(third)-4x(squared)(2)(x-2)(1)/(x-2)(fourth)

    than

    (x-2)[(3x(squared)-8x)(x-2)-2x(third)+8x(squared)]/(x-2)(fourth)

    now stuck

    any help on further factoring would be appreciated
    F(x) = \frac{x^3 - 4x^2}{(x - 2)^2}

    So
    F^{\prime}(x) = \frac{(3x^2 - 8x)(x - 2)^2 - (x^3 - 4x^2)(2(x - 2))}{(x - 2)^4}

    F^{\prime}(x) = \frac{(x - 2)[(3x^2 - 8x)(x - 2) - 2(x^3 - 4x^2)]}{(x - 2)^4}

    This is about where you got to. Now divide through by the common x - 2:
    F^{\prime}(x) = \frac{(3x^2 - 8x)(x - 2) - 2(x^3 - 4x^2)}{(x - 2)^3}

    Expand:
    F^{\prime}(x) = \frac{3x^3 - 8x^2 - 6x^2 + 16x - 2x^3 + 8x^2)}{(x - 2)^3}

    And simplify:
    F^{\prime}(x) = \frac{x^3 - 6x + 16}{(x - 2)^3}

    -Dan
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  3. Nov 18th 2007, 08:05 AM #3
    Soroban
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    Hello, BClary!

    You seem to be doing fine . . .

    Find the first and second derivatives: . F(x) \;=\;\frac{x^3-4x^2}{(x-2)^2}

    Quotient Rule: . F'(x) \;=\;\frac{(x-2)^2\cdot(3x^2-8x) - (x^3-4x^2)\cdot2(x-2)}{(x-2)^4}

    . . . . . . . . . . . F'(x) \;=\;\frac{x(x-2)^2(3x-8) - 2x^2(x-4)(x-2)} {(x-2)^4}

    Factor/reduce: . F'(x) \;=\;\frac{x(x-2)\cdot\left[(x-2)(3x-8) - 2x(x-4)\right]}{(x-2)^4} \;= . \frac{x\cdot\left[(x-2)(3x-8) - 2x(x-4)\right]}{(x-2)^3}

    Simplify: . . . . F'(x) \;=\;\frac{x\left[3x^2 - 8x - 6x + 16 - 2x^2 + 8x\right]}{(x-2)^3}

    Therefore: . . . \boxed{F'(x) \;=\;\frac{x\left(x^2-6x+16\right)}{(x-2)^3}}


    Now try the second derivative . . .
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  4. Nov 18th 2007, 08:09 AM #4
    BClary
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    Thanks to all, off doing second now

    Thanks for all your help, lets see where I end up at

    thanks
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  5. Nov 18th 2007, 08:57 AM #5
    BClary
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    2nd derivative should be

    after a little algrebra (haha)

    (-8x-32)/((x-2)(fourth))

    Right?
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  6. Nov 18th 2007, 08:59 AM #6
    topsquark
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    Quote Originally Posted by BClary View Post
    after a little algrebra (haha)

    (-8x-32)/((x-2)(fourth))

    Right?
    Yup!

    -Dan
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  7. Nov 18th 2007, 09:00 AM #7
    BClary
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    Thanks so much

    TY Ty Ty Ty ty TY
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