\(\frac{1}{\sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
21.9
- Applied taylor to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}\]
21.9
- Taylor expanded around 0 to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{x}^{\frac{1}{3}}}\]
21.9
- Applied simplify to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}}\]
14.2
- Using strategy
rm 14.2
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}}\]
14.2
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
14.2
- Using strategy
rm 14.2
- Applied add-cube-cbrt to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}{\color{blue}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\right)}^3}}\]
14.4
- Applied add-cube-cbrt to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}\right)}^3}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\right)}^3}\]
14.4
- Applied cube-undiv to get
\[\color{red}{\frac{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}\right)}^3}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}}\right)}^3}\]
14.4
- Applied taylor to get
\[{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{\sqrt[3]{{\left({\left(1 + x\right)}^2\right)}^{\frac{1}{3}} + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(1 + x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x}\right)}}\right)}^3\]
14.4
- Taylor expanded around 0 to get
\[\color{red}{{\left(\frac{\sqrt[3]{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{\sqrt[3]{{\left({\left(1 + x\right)}^2\right)}^{\frac{1}{3}} + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(1 + x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x}\right)}}\right)}^3} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{\sqrt[3]{{\left({\left(1 + x\right)}^2\right)}^{\frac{1}{3}} + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(1 + x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x}\right)}}\right)}^3}\]
14.4
- Applied simplify to get
\[{\left(\frac{\sqrt[3]{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - x}}{\sqrt[3]{{\left({\left(1 + x\right)}^2\right)}^{\frac{1}{3}} + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(1 + x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x}\right)}}\right)}^3 \leadsto \frac{\left(1 + x\right) - x}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)}}\]
12.0
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(1 + x\right) - x}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)}}} \leadsto \color{blue}{\frac{1}{\sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}}\]
7.4
