Question

Find f'(x) and F"(x). f(x)=9+ 3x – 3x^3

Answer 1

`f'(x)=3-9x^(2)` and
`f''(x)=-18x`

Explanation:

In order to find the derivatives, first we need to remember that for polynomial functions:

`f'(x)=(x^(n)+x^(m))'= (x^(n))'+(x^(m))'`, as well as that:

`f'(x)= (x^(n))' = n*(x^(n-1))`

1. First derivative of the function:

`f(x)=9+3x-3x^(3)`

`f'(x)=(9)'+(3x)'-(3x^(3))'` using the property
`f'(x)= (x^(n))' = n*(x^(n-1))` then

`f'(x)=3-3*3x^(2)`, remember that the derivative of a constant is equal to 0

`f'(x)=3-9x^(2)`

2. Second derivative:

`f'(x)=3-9x^(2)`

`f''(x)=(3-9x^(2))'` using the property
`f'(x)= (x^(n))' = n*(x^(n-1))` then

`f''(x)=(3)'-(9x^(2))'`

`f''(x)=-(9*2)x^(1)`

`f''(x)=-18x`

In conclusion,
`f'(x)=3-9x^(2)` and
`f''(x)=-18x`