\(\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^2}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
40.6
- Using strategy
rm 40.6
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
40.5
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
40.5
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
62.1
- Taylor expanded around -inf to get
\[\frac{\color{red}{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
62.1
- Applied simplify to get
\[\color{red}{\frac{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}}\]
22.8
- Applied taylor to get
\[\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left({x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}\]
22.8
- Taylor expanded around 0 to get
\[\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left(\color{red}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left(\color{blue}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}\]
22.8
- Applied simplify to get
\[\color{red}{\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{\left({x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}}} \leadsto \color{blue}{\frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}}\]
42.2
- Applied taylor to get
\[\frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2} \leadsto \frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\left({x}^{\frac{1}{3}}\right)}^2}\]
42.2
- Taylor expanded around 0 to get
\[\frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\color{red}{\left({x}^{\frac{1}{3}}\right)}}^2} \leadsto \frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\color{blue}{\left({x}^{\frac{1}{3}}\right)}}^2}\]
42.2
- Applied simplify to get
\[\frac{\left(1 - \frac{1}{x}\right) - \frac{-1}{x}}{\left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {\left({x}^{\frac{1}{3}}\right)}^2} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^2}\]
1.7
- Applied final simplification
