Let's start off with the definition of the quotient rule:
d/dx (f(x)/(g(x))) = (g(x)f'(x) - f(x)g'(x))/(g(x))^2
Or to make remembering the rule simpler, let f(x) be "high," being the upper function and let g(x) be "low," being the lower function to give us:
d/dx ((high)/(low)) = (low*d'(high) - (high*d'(low)))/(low)^2
where d' denotes the "derivative of."
This formula can be read as "low d high minus high d low over low squared," which has a nice flow for memorization.
In the case f(x) = (1-x)/(x^3 - 6), let "low" denote x^3 - 6 and "high" denote 1-x
Applying the quotient rule,
d/dx ((high)/(low)) = ((x^3 - 6)*d'(1-x) - ((1-x)*d'(x^3 - 6)))/(x^3 - 6)^2
Now switching ((high)/(low)) to f(x) for simplicity,
d/dx (f(x)) = ((x^3 - 6)⋅(-1) - ((1-x)*(3x^2)))/(x^3 - 6)^2
Knowing that by the power rule d/dx (x^n) = nx^(n-1)
And that the derivative of a constant is 0.
d/dx (f(x)) = ((-x^3 + 6) - ((3x^2-3x^3)))/(x^3 - 6)^2
d/dx (f(x)) = (-x^3 + 6 - 3x^2+3x^3)/(x^3 - 6)^2 = (2x^3-3x^2+6)/((x^3 - 6)^2)