Question #8b3ee

1 Answer
Oct 12, 2017

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