\({\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{{x}^2 \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\right)}^3\)
- Started with
\[\frac{e^{x}}{e^{x} - 1}\]
18.3
- Applied taylor to get
\[\frac{e^{x}}{e^{x} - 1} \leadsto \frac{e^{x}}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
5.2
- Taylor expanded around 0 to get
\[\frac{e^{x}}{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
5.2
- Using strategy
rm 5.2
- Applied add-cube-cbrt to get
\[\frac{e^{x}}{\color{red}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}} \leadsto \frac{e^{x}}{\color{blue}{{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}^3}}\]
5.6
- Applied add-cube-cbrt to get
\[\frac{\color{red}{e^{x}}}{{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{e^{x}}\right)}^3}}{{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}^3}\]
5.7
- Applied cube-undiv to get
\[\color{red}{\frac{{\left(\sqrt[3]{e^{x}}\right)}^3}{{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\right)}^3}\]
5.7
- Applied simplify to get
\[{\color{red}{\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{e^{x}}}{\sqrt[3]{{x}^2 \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\right)}}^3\]
0.6
