Initial program 29.9
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.7
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]
Taylor expanded around inf 0.2
\[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
- Using strategy
rm Applied add-sqr-sqrt0.4
\[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
- Using strategy
rm Applied pow1/20.4
\[\leadsto \frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}}\]
Applied pow1/20.4
\[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}\]
Applied pow-prod-down0.2
\[\leadsto \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\frac{1}{2}}}}\]
Applied pow-flip0.2
\[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}}\]
Simplified0.2
\[\leadsto {\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\color{blue}{\frac{-1}{2}}}\]
Final simplification0.2
\[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
