<s>[INST] How can I simplify the derivative further to match the correct answer? I've been stuck on trying to simplify my expression in order to match the correct answer but I can't seem to get the correct solution. Guidance towards the proper steps would be greatly appreciated!
The original question:

Differentiate and simplify completely
  $$g(x)=\sqrt[3]{\frac{x^3-1}{x^3+1}}$$

My solution:

$$g^\prime(x) = \frac{2x^2}{\sqrt[3]{\left(x^3-1\right)^2}\sqrt[3]{\left(x^3+1\right)^4}}$$

I need help simplifying this to become this solution:

$$g^\prime(x) = \frac{2x^2\;\sqrt[3]{\left(x^3+1\right)^2}\;\sqrt[3]{x^3-1}}{(x+1)^2\left(x^2-x+1\right)^2(x-1)\left(x^2+x+1\right)} $$

Original image of solution.
 [/INST] Just multiply by $1$ (to free the denominator from irrationality):
$$g^\prime(x) = \frac{2x^2}{\sqrt[3]{\left(x^3-1\right)^2}\sqrt[3]{\left(x^3+1\right)^4}}\cdot \frac{\sqrt[3]{\left(x^3+1\right)^2}\;\sqrt[3]{x^3-1}}{\sqrt[3]{\left(x^3+1\right)^2}\;\sqrt[3]{x^3-1}}=\\
\frac{2x^2\;\sqrt[3]{\left(x^3+1\right)^2}\;\sqrt[3]{x^3-1}}{\color{red}{(x^3+1)^2}\color{blue}{(x^3-1)}}=\\
\frac{2x^2\;\sqrt[3]{\left(x^3+1\right)^2}\;\sqrt[3]{x^3-1}}{\color{red}{(x+1)^2\left(x^2-x+1\right)^2}\color{blue}{(x-1)\left(x^2+x+1\right)}}.$$
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