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Math Help - exponential differentiation.

  1. #1
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    exponential differentiation.

    Find the coordinates of the turning point on the graph of  y = x\times e^{2x} and determine its
    nature.

    I used the product rule and got  e^{2x} +2x\times e^{2x}

    so to find the coordinates of the turning point I set that first derivative equal to 0.

     e^{2x} +2x\times e^{2x} = 0

     e^{2x} (1+2x) = 0

     e^{2x} = 0 or  1+2x = 0

     x = -\frac{1}{2}

    and to find the 'nature' of the turning points I have to take the second derivative. in which I dont get the correct answer, heres my working,

     \frac{dy}{dx} = e^{2x} + 2xe^{2x}

    the derivative of  e^{2x} = 2e^{2x}
    using the product rule on  2x e^{2x}

     2 \times e^{2x} + 2e^{2x} \times 2x

     \frac{d^{2}y}{dx^{2}} = 2e^{2x} + 2 \times e^{2x} + 2e^{2x} \times 2x

    =  4e^{2x} (1+x)

    but the my book says that the correct answer is   \frac{d^{2}y}{dx^{2}} =  2(2x+1)e^{2x} + e^{2x}.2
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  2. #2
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    Quote Originally Posted by Tweety View Post
    Find the coordinates of the turning point on the graph of  y = x\times e^{2x} and determine its
    nature.

    I used the product rule and got  e^{2x} +2x\times e^{2x}

    so to find the coordinates of the turning point I set that first derivative equal to 0.

     e^{2x} +2x\times e^{2x} = 0

     e^{2x} (1+2x) = 0

     e^{2x} = 0 or  1+2x = 0

     x = -\frac{1}{2}

    and to find the 'nature' of the turning points I have to take the second derivative. in which I dont get the correct answer, heres my working,

     \frac{dy}{dx} = e^{2x} + 2xe^{2x}

    the derivative of  e^{2x} = 2e^{2x}
    using the product rule on  2x e^{2x}

     2 \times e^{2x} + 2e^{2x} \times 2x

     \frac{d^{2}y}{dx^{2}} = 2e^{2x} + 2 \times e^{2x} + 2e^{2x} \times 2x

    =  4e^{2x} (1+x)

    but the my book says that the correct answer is  \frac{d^{2}y}{dx^{2}} = 2(2x+1)e^{2x} + e^{2x}.2

    \frac{d(2xe^{2x})}{dx} = 2e^{2x}+4xe^{2x}

    \frac{d(e^{2x})}{dx}=2e^{2x}

    Sum the above and you get the 2nd derivative is 2(2x+2)e^{2x}=2(2x+1)e^{2x}+2e^{2x}

    Tonio

    Ps. Oh, and PLEASE: do not use X to denote multiplication...this is not elementary school!
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