- Started with
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
21.4
- Using strategy
rm 21.4
- Applied div-sub to get
\[wj - \color{red}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
21.4
- Applied simplify to get
\[wj - \left(\color{red}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \leadsto wj - \left(\color{blue}{\frac{wj}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
21.3
- Applied simplify to get
\[wj - \left(\frac{wj}{wj + 1} - \color{red}{\frac{x}{e^{wj} + wj \cdot e^{wj}}}\right) \leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{1 + wj}}{e^{wj}}}\right)\]
21.3
- Applied taylor to get
\[wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) \leadsto wj - \left(\left(\left({wj}^{3} + wj\right) - {wj}^2\right) - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\]
12.0
- Taylor expanded around 0 to get
\[wj - \left(\color{red}{\left(\left({wj}^{3} + wj\right) - {wj}^2\right)} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) \leadsto wj - \left(\color{blue}{\left(\left({wj}^{3} + wj\right) - {wj}^2\right)} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\]
12.0
- Applied simplify to get
\[wj - \left(\left(\left({wj}^{3} + wj\right) - {wj}^2\right) - \frac{\frac{x}{1 + wj}}{e^{wj}}\right) \leadsto \left(\left(wj - {wj}^3\right) - \left(wj - {wj}^2\right)\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\]
6.4
- Applied final simplification
- Applied simplify to get
\[\color{red}{\left(\left(wj - {wj}^3\right) - \left(wj - {wj}^2\right)\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}} \leadsto \color{blue}{\left({wj}^2 - {wj}^3\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}}\]
0.1
