Initial program 30.0
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.8
\[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]
- Using strategy
rm Applied add-log-exp31.7
\[\leadsto \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \color{blue}{\log \left(e^{\sqrt{x} \cdot \sqrt{x}}\right)}}{\sqrt{x + 1} + \sqrt{x}}\]
Applied add-log-exp31.1
\[\leadsto \frac{\color{blue}{\log \left(e^{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right)} - \log \left(e^{\sqrt{x} \cdot \sqrt{x}}\right)}{\sqrt{x + 1} + \sqrt{x}}\]
Applied diff-log31.1
\[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{e^{\sqrt{x} \cdot \sqrt{x}}}\right)}}{\sqrt{x + 1} + \sqrt{x}}\]
Simplified29.4
\[\leadsto \frac{\log \color{blue}{\left(e^{\left(1 + x\right) - x}\right)}}{\sqrt{x + 1} + \sqrt{x}}\]
- Using strategy
rm Applied associate--l+0.2
\[\leadsto \frac{\log \left(e^{\color{blue}{1 + \left(x - x\right)}}\right)}{\sqrt{x + 1} + \sqrt{x}}\]
- Using strategy
rm Applied add-sqr-sqrt0.3
\[\leadsto \frac{\log \left(e^{1 + \left(x - x\right)}\right)}{\color{blue}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
Applied add-cube-cbrt0.3
\[\leadsto \frac{\log \left(e^{\color{blue}{\left(\sqrt[3]{1 + \left(x - x\right)} \cdot \sqrt[3]{1 + \left(x - x\right)}\right) \cdot \sqrt[3]{1 + \left(x - x\right)}}}\right)}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Applied exp-prod0.3
\[\leadsto \frac{\log \color{blue}{\left({\left(e^{\sqrt[3]{1 + \left(x - x\right)} \cdot \sqrt[3]{1 + \left(x - x\right)}}\right)}^{\left(\sqrt[3]{1 + \left(x - x\right)}\right)}\right)}}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Applied log-pow0.3
\[\leadsto \frac{\color{blue}{\sqrt[3]{1 + \left(x - x\right)} \cdot \log \left(e^{\sqrt[3]{1 + \left(x - x\right)} \cdot \sqrt[3]{1 + \left(x - x\right)}}\right)}}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Applied times-frac0.4
\[\leadsto \color{blue}{\frac{\sqrt[3]{1 + \left(x - x\right)}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\log \left(e^{\sqrt[3]{1 + \left(x - x\right)} \cdot \sqrt[3]{1 + \left(x - x\right)}}\right)}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}\]
Simplified0.3
\[\leadsto \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(-\frac{1}{2}\right)}} \cdot \frac{\log \left(e^{\sqrt[3]{1 + \left(x - x\right)} \cdot \sqrt[3]{1 + \left(x - x\right)}}\right)}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
Simplified0.3
\[\leadsto {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\left(-\frac{1}{2}\right)}}\]
Final simplification0.3
\[\leadsto {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(-\frac{1}{2}\right)}\]
