\(\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\left(\frac{1}{3}\right)}}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
29.8
- Using strategy
rm 29.8
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
29.7
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
29.7
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
32.7
- Taylor expanded around inf to get
\[\frac{\color{red}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
32.7
- Applied simplify to get
\[\frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right)}\]
32.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\left(\frac{1}{3}\right)}}}\]
3.1
