Question

How do you find the derivative for `f(x) = 1 / [root(3)(3 - x^3)]`?

Answer 1

We will need to use the chain rule plus some basic properties of exponents and simple derivatives for this.

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

Let `h(x) = (3-x^3)`
and `g(x) = x^(-1/3)`

so `f(x) = g(h(x))`

Note that `(df(x))/(dx) = (d g(h(x)))/(d h(x)) * (d h(x))/(dx)`

`(d g(h(x)))/(d h(x)) = -1/3(h(x))^(-4/3)`

`(d h(x))/(dx) = -3x^2`

`(d f(x))/(dx) = (-1/3(h(x))^(-4/3))*(-3x^2)`

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

`= (x^2)/((3-x^3)root(3)(3-x^3))`