\(\frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
52.5
- Using strategy
rm 52.5
- Applied add-log-exp to get
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
52.6
- Applied add-log-exp to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
52.6
- Applied diff-log to get
\[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
52.6
- Applied simplify to get
\[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
52.5
- Using strategy
rm 52.5
- Applied flip-- to get
\[\log \left(e^{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)\]
52.6
- Applied taylor to get
\[\log \left(e^{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
44.1
- Taylor expanded around inf to get
\[\log \left(e^{\frac{\color{red}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{\color{blue}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
44.1
- Applied simplify to get
\[\log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\]
23.7
- Applied final simplification
