\(\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
18.6
- Applied taylor to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}\]
18.6
- Taylor expanded around 0 to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{x}^{\frac{1}{3}}}\]
18.6
- Applied simplify to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}}\]
8.4
- Using strategy
rm 8.4
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}}\]
8.4
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
8.4
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
8.4
- Taylor expanded around 0 to get
\[\frac{\color{red}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
8.4
- Applied simplify to get
\[\frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{1 + \left(x - x\right)}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}\]
0.9
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{1 + \left(x - x\right)}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}}\]
0.9
