\(\frac{(\left({x}^2\right) * 3 + \left((x * 4 + 2)_*\right))_*}{e^{(x * 1 + x)_*} \cdot 2}\)
- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
18.8
- Using strategy
rm 18.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
18.8
- Applied un-div-inv to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
18.8
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
18.7
- Applied un-div-inv to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
18.8
- Applied frac-sub to get
\[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
23.7
- Applied simplify to get
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{red}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{blue}{e^{(x * \left(1 + 0\right) + x)_*}}}}{2}\]
23.7
- Applied taylor to get
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2}\]
5.3
- Taylor expanded around 0 to get
\[\frac{\frac{\color{red}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{\frac{\color{blue}{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}}{e^{(x * \left(1 + 0\right) + x)_*}}}{2}\]
5.3
- Applied simplify to get
\[\frac{\frac{3 \cdot {x}^2 + \left(2 + 4 \cdot x\right)}{e^{(x * \left(1 + 0\right) + x)_*}}}{2} \leadsto \frac{(\left(x \cdot x\right) * 3 + \left((x * 4 + 2)_*\right))_*}{2 \cdot e^{(x * 1 + x)_*}}\]
5.3
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{(\left(x \cdot x\right) * 3 + \left((x * 4 + 2)_*\right))_*}{2 \cdot e^{(x * 1 + x)_*}}} \leadsto \color{blue}{\frac{(\left({x}^2\right) * 3 + \left((x * 4 + 2)_*\right))_*}{e^{(x * 1 + x)_*} \cdot 2}}\]
5.3
