\(\frac{\left({\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}\)
- Started with
\[e^{a \cdot x} - 1\]
22.7
- Applied taylor to get
\[e^{a \cdot x} - 1 \leadsto \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1\]
19.4
- Taylor expanded around 0 to get
\[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1 \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right)} - 1\]
19.4
- Applied simplify to get
\[\color{red}{\left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + a \cdot x\right)\right) - 1} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a}\]
0.2
- Using strategy
rm 0.2
- Applied distribute-lft1-in to get
\[\color{red}{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right) + x \cdot a} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right) \cdot \left(x \cdot a\right)}\]
0.2
- Using strategy
rm 0.2
- Applied flip-+ to get
\[\color{red}{\left(\frac{1}{2} \cdot \left(x \cdot a\right) + 1\right)} \cdot \left(x \cdot a\right) \leadsto \color{blue}{\frac{{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}} \cdot \left(x \cdot a\right)\]
0.2
- Applied associate-*l/ to get
\[\color{red}{\frac{{\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1} \cdot \left(x \cdot a\right)} \leadsto \color{blue}{\frac{\left({\left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}^2 - {1}^2\right) \cdot \left(x \cdot a\right)}{\frac{1}{2} \cdot \left(x \cdot a\right) - 1}}\]
2.6
