Initial program 19.5
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub19.4
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Applied simplify19.4
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- 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
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Applied simplify19.3
\[\leadsto \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\color{blue}{(\left(\sqrt{x + 1}\right) \cdot x + \left((x \cdot \left(\sqrt{x}\right) + \left(\sqrt{x}\right))_*\right))_*}}\]
- Using strategy
rm Applied add-sqr-sqrt19.4
\[\leadsto \color{blue}{\sqrt{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{(\left(\sqrt{x + 1}\right) \cdot x + \left((x \cdot \left(\sqrt{x}\right) + \left(\sqrt{x}\right))_*\right))_*}} \cdot \sqrt{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{(\left(\sqrt{x + 1}\right) \cdot x + \left((x \cdot \left(\sqrt{x}\right) + \left(\sqrt{x}\right))_*\right))_*}}}\]
Applied simplify19.4
\[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{1 + x}}}{(\left(\sqrt{x}\right) \cdot \left(\sqrt{1 + x}\right) + x)_*}}} \cdot \sqrt{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{(\left(\sqrt{x + 1}\right) \cdot x + \left((x \cdot \left(\sqrt{x}\right) + \left(\sqrt{x}\right))_*\right))_*}}\]
Applied simplify0.5
\[\leadsto \sqrt{\frac{\frac{1}{\sqrt{1 + x}}}{(\left(\sqrt{x}\right) \cdot \left(\sqrt{1 + x}\right) + x)_*}} \cdot \color{blue}{\sqrt{\frac{\frac{1}{\sqrt{1 + x}}}{(\left(\sqrt{x}\right) \cdot \left(\sqrt{1 + x}\right) + x)_*}}}\]
