Initial program 19.5
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub19.5
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified19.5
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm Applied flip--19.3
\[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/19.3
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Simplified0.8
\[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
- Using strategy
rm Applied associate-/r*0.4
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\color{blue}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Applied div-inv0.4
\[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
Applied times-frac0.4
\[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
Simplified0.4
\[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}\]
Simplified0.3
\[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{(\left(\sqrt{x}\right) \cdot \left(\sqrt{x + 1}\right) + \left(x + 1\right))_*}}\]
Final simplification0.3
\[\leadsto \frac{\frac{1}{\sqrt{x}}}{(\left(\sqrt{x}\right) \cdot \left(\sqrt{x + 1}\right) + \left(x + 1\right))_*}\]
