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

Question

1949 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-27T20:33:51

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))#