Initial program 29.5
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
- Using strategy
rm Applied add-exp-log29.5
\[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}}\]
- Using strategy
rm Applied flip3--29.4
\[\leadsto e^{\log \color{blue}{\left(\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)}}\]
Applied log-div29.4
\[\leadsto e^{\color{blue}{\log \left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) - \log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}}\]
Applied exp-diff29.4
\[\leadsto \color{blue}{\frac{e^{\log \left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right)}}{e^{\log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}}}\]
Simplified2.6
\[\leadsto \frac{\color{blue}{1}}{e^{\log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}}\]
- Using strategy
rm Applied rem-exp-log0.5
\[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}}\]
Taylor expanded around 0 31.9
\[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \color{blue}{{x}^{\frac{1}{3}}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]
Simplified0.5
\[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]
Final simplification0.5
\[\leadsto \frac{1}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]
