\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]

\[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]

\[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]

\[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]

\[\color{red}{e^{\frac{\log_* (1 + x)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]

\[\frac{\color{red}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]

\[\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\frac{1}{3}}\right)}\]

\[\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot \color{red}{{x}^{\frac{1}{3}}}\right)} \leadsto \frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)}\]

\[\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\frac{1}{3}}\right)} \leadsto \frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{(\left(\sqrt[3]{x}\right) * \left(e^{\frac{\log_* (1 + x)}{3}}\right) + \left((\left(e^{\frac{\log_* (1 + x)}{3}}\right) * \left(e^{\frac{\log_* (1 + x)}{3}}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*\right))_*}\]