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

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### 1 Answer

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`