Initial program 19.6
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm
Applied frac-sub 19.6
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Applied simplify 19.6
\[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
- Using strategy
rm
Applied flip-- 19.3
\[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Applied associate-/l/ 19.3
\[\leadsto \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]
Applied simplify 19.3
\[\leadsto \frac{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\color{blue}{x \cdot \sqrt{x + 1} + \left(x + 1\right) \cdot \sqrt{x}}}\]
Applied taylor 19.3
\[\leadsto \frac{\left(1 + x\right) - {\left(\sqrt{x}\right)}^2}{x \cdot \sqrt{x + 1} + \left(x + 1\right) \cdot \sqrt{x}}\]
Taylor expanded around 0 19.3
\[\leadsto \frac{\color{blue}{\left(1 + x\right)} - {\left(\sqrt{x}\right)}^2}{x \cdot \sqrt{x + 1} + \left(x + 1\right) \cdot \sqrt{x}}\]
Applied simplify 0.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{x + 1} \cdot x + \sqrt{x} \cdot \left(x + 1\right)}}\]
- Removed slow pow expressions
