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Math Help - differentiating a rational quadratic

  1. #1
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    differentiating a rational quadratic

    Hello, I am trying to differentiate this function so I work out where it is increasing or decreasing.
    What is the best procedure to do this. I tried using quotient rule but it seemed very laborious. Is there a better way ?

    2x^2+x-1 / x^2+x-2

    Thanks for any help.
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  2. #2
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    Re: differentiating a rational quadratic

    Maybe try factoring each quadratic with the purpose of cancellation. Otherwise your stuck with the quotient rule.
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    Re: differentiating a rational quadratic

    So our function is, f(x)=\frac{2x^2+x-1}{x^2+x-2} .

    You need to find f'(x).

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


     =\frac{(4x+1)(x^2+x-2)-(2x+1)(2x^2+x-1)}{(x^2+x-2)^2}

    Try to continue.
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    Re: differentiating a rational quadratic

    thank you. I have tried to continue to simplify this derived function so that I can work out it's roots.
    I can see I could split it and cancel one of the (x^2+x-2) factors, but that is as far as I can get.
    I really am stuck with this one. Any help proceeding would be great.
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  5. #5
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    Re: differentiating a rational quadratic

     =\frac{(4x+1)(x^2+x-2)-(2x+1)(2x^2+x-1)}{(x^2+x-2)^2}
    as stated by AsZ, expand the numerator and combine like terms ... in other words, do the grunt work algebra and finish simplifying the numerator.
    You'll end up with a quadratic in the numerator that won't factor, so you'll need to complete the square or use the quadratic formula to find where the derivative equals 0.
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    Re: differentiating a rational quadratic

    my grunt work gives me:
    x^2-6x-1 / x^4+2x^3-3x^2-4x+4
    I equated the numerator to zero to find the roots. That gave me roots at 6.16 and -0.162 however looking at a graph of 2x^2+x-1 / x^2+x-2 that cannot be correct.
    I double checked but to no avail.
    If anyone can spot my error that would be appreciated.
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    Re: differentiating a rational quadratic

    Quote Originally Posted by fran1942 View Post
    my grunt work gives me:
    x^2-6x-1 / x^4+2x^3-3x^2-4x+4
    I equated the numerator to zero to find the roots. That gave me roots at 6.16 and -0.162 however looking at a graph of 2x^2+x-1 / x^2+x-2 that cannot be correct.
    I double checked but to no avail.
    If anyone can spot my error that would be appreciated.
    your roots for the derivative are correct, x = 3 \pm \sqrt{10} ... note the relative max at x = 3-\sqrt{10} and the relative min at x = 3+\sqrt{10} in the graph (marked by the red diamonds)
    Attached Thumbnails Attached Thumbnails differentiating a rational quadratic-rational.jpg  
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