Question

Question #8b3ee

Answer 1

`f'(x) = (-9x^2)/(x^3-4)^2`

Explanation:

There are two ways to do this, each of which can get you to the correct solution.

Quotient Rule

If `f(x) = (g(x))/(h(x))`, then `f'(x) = ( h(x)*g'(x) - g(x)*h'(x) ) / ( (h(x))^2)`

`f'(x) = ( (x^3-4)(0) - (3)(3x^2) ) / (x^3-4)^2 = (-9x^2) / (x^3-4)^2`

Chain Rule

If `f(x) = g(h(x))`, then `f'(x) = g'(h(x))*h'(x)`

`f(x) = 3/(x^3-4) = 3(x^3-4)^-1`

If one considers that `g(x) = 3x^-1` and `h(x) = x^3-4`, then:

`f'(x) = (-1) * 3(x^3-4)^(-2) * (3x^2) = -9x^2(x^3-4)^(-2)`

`= (-9x^2)/(x^3-4)^2`