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Determine the second derivative of the function `f(x)=(x^2+9)^4`

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user8724300 | eNotes Newbie

Posted April 1, 2013 at 7:36 PM via web

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Determine the second derivative of the function

`f(x)=(x^2+9)^4`

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted April 1, 2013 at 7:54 PM (Answer #1)

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Here you need to use chain rule:

`(f(g(x)))'=f'(g(x))cdot g'(x)`

First derivative:

`f'(x)=4(x^2+9)^3 2x=8x(x^2+9)^3`

Second derivative is derivative of the first derivative.

We will need to us product rule:

`(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)`

So we have

`f''(x)=8(x^2+9)^3+8x(3(x^2+9)^2 2x)`

`f''(x)=8(x^2+9)^3+48x^2(x^2+9)^2`

So second derivative of function `f` is

`f''(x)=8(x^2+9)^3+48x^2(x^2+9)^2`

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