\begin{align*}
  \left(x^{1/3}-x\right) \cdot \left(x^{2/3}+x^2\right)
  &= x^{1/3} \cdot x^{2/3} + x^{1/3} \cdot x^2 - x\cdot x^{2/3} - x \cdot x^{2} \\
  &= x + x^{7/3} - x^{5/3} - x^3 \\
\end{align*}


\begin{align*}
  \frac{d}{dx} \left( x + x^{7/3} - x^{5/3} - x^3 \right)
  &= 1 + \frac{7}{3}x^{4/3} - \frac{5}{3}x^{2/3} - 3x^2 \\
  &=  - 3x^2 - \frac{5}{3}x^{2/3} + \frac{7}{3}x^{4/3} + 1 
\end{align*}