Question

How do you differentiate `f(x)= -1 / (x^3-4 )` using the quotient rule?

Answer 1

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

Explanation:

The quotient rule is stated thus:

If `f(x)=u/v`
then `f'(x) = (v(du)/dx-u(dv)/dx)/v^2`

Let's apply that rule to our function : `f(x)= -1/(x^3-4)`

Step 1:
`v(du)/dx = (x^3-4)(0)=0`

Step 2:
`u(dv)/dx= (-1)(3x^2)=-3x^2`

Step 3:
`v^2= (x^3-4)^2`

Step 4:
`f'(x)= (v(du)/dx-u(dv)/dx)/v^2 = (0-(-3x^2))/(x^3-4)^2 = color(blue)((3x^2)/(x^3-4)^2)`