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Math Help - chain rule second derivative

  1. #1
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    chain rule second derivative

    I'm supposed to find the second derivative of (x^2+5)^8
    What I got was: f(u)=u^8, u(x)=(x^2+5)
    (8(x^2+5)^7)(2)=16(x^2+5)^7
    f(u)=16(u)^7, u(x)=(x^2+5)
    (16)7(x^2+5)^6(2)=224(x^2+5)^6
    but the back of the book says it's: 80(x^2+5)^6(3x^2+1)
    Any help would be much appreciated!
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  2. #2
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    f = (x^2 + 5)^8.

    Let u = x^2 + 5 so that f = u^8.


    \frac{du}{dx} = 2x.

    \frac{df}{du} = 8u^7 = 8(x^2 + 5)^7.


    Therefore \frac{df}{dx} = 16x(x^2 + 5)^7.


    Now you will need to use the product rule...

    \frac{d^2f}{dx^2} = 16x\cdot 2x \cdot 7(x^2 + 5)^6 + 16(x^2 + 5)^7

     = 224x^2(x^2 + 5)^6 + 16(x^2 + 5)^7

     = 16(x^2 + 5)^6(14x^2 + x^2 + 5)

     = 16(x^2 + 5)^6(15x^2 + 5)

     = 80(x^2 + 5)^6(3x^2 + 1).
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  3. #3
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    oh duh, 2x not 2. thanks!
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