\(\frac{\frac{\left(2 + \frac{2}{\varepsilon}\right) + \frac{x \cdot 2}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}{2 + \frac{2}{\varepsilon}}\)
- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
39.7
- Using strategy
rm 39.7
- Applied flip-- to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
45.5
- Applied associate-*l/ to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2}\]
45.5
- Applied neg-sub0 to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{red}{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{0 - \left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
45.5
- Applied exp-diff to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{0 - \left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
45.5
- Applied associate-*r/ to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
45.5
- Applied frac-sub to get
\[\frac{\color{red}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{0}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}}{2}\]
51.2
- Applied simplify to get
\[\frac{\frac{\color{red}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{0}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{{\left(1 + \frac{1}{\varepsilon}\right)}^2 - \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}{e^{x \cdot \varepsilon}} \cdot \frac{{\left(\frac{1}{\varepsilon}\right)}^2 - 1}{e^{x}}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
51.2
- Applied taylor to get
\[\frac{\frac{{\left(1 + \frac{1}{\varepsilon}\right)}^2 - \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}{e^{x \cdot \varepsilon}} \cdot \frac{{\left(\frac{1}{\varepsilon}\right)}^2 - 1}{e^{x}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
5.7
- Taylor expanded around 0 to get
\[\frac{\frac{\color{red}{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
5.7
- Applied simplify to get
\[\frac{\frac{2 + \left(2 \cdot \frac{x}{\varepsilon} + 2 \cdot \frac{1}{\varepsilon}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\left(2 + \frac{2}{\varepsilon}\right) + \frac{x \cdot 2}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}{2 + \frac{2}{\varepsilon}}\]
11.3
- Applied final simplification
