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Math Help - Calculating the second derivative.

  1. #1
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    Calculating the second derivative.

    \frac{x}{x+5}

    I have an answer in my solutions manual that I think might be a typo. It is my understanding right now that the second derivative of x is 0. So if that is correct then the second derivative of the equation at the top of my post should then be:

    \frac{0}{0}

    I want to make sure I have not made an error.

    My answer:
    0

    Thank in advance...
    Last edited by sepoto; November 10th 2013 at 12:58 PM.
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  2. #2
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    Re: Calculating the second derivative.

    Quote Originally Posted by sepoto View Post
    \frac{x}{x+5}

    I have an answer in my solutions manual that I think might be a typo. It is my understanding right now that the second derivative of x is 0. So if that is correct then the second derivative of the equation at the top of my post should then be:

    \frac{0}{0}

    I want to make sure I have not made an error.

    My answer:
    0

    Thank in advance...
    Yes, you made an error. You can apply the quotient rule. You can apply the product rule to x(x+5)^{-1}. But, you cannot take the derivative of the numerator, then the derivative of the denominator, and assume you are getting a correct derivative.

    You can also simplify the expression: \dfrac{x}{x+5} = \dfrac{x+(5-5)}{x+5} = \dfrac{x+5}{x+5} - \dfrac{5}{x+5} = 1-\dfrac{5}{x+5} = 1-5(x+5)^{-1}

    Then, the derivative would be 0-5(-1)(x+5)^{-2} = 5(x+5)^{-2} and the second derivative would be -10(x+5)^{-3}.
    Thanks from sepoto
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  3. #3
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    Re: Calculating the second derivative.

    Note that I am assuming you know the chain rule. If you do not know the chain rule, then you cannot take the derivative of (x+5)^{-1}. Instead, you would need to use the quotient rule:

    \dfrac{d}{dx}\dfrac{x}{x+5} = \dfrac{(x+5)\tfrac{d}{dx}(x) - x\tfrac{d}{dx}(x+5)}{(x+5)^2} = \dfrac{(x+5)\cdot 1 - x\cdot 1}{(x+5)^2} = \dfrac{5}{x^2+10x+25}

    Then apply the quotient rule again to find the second derivative.
    Thanks from sepoto
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