Solve alpha/beta+1+beta/alpha+1

$$\frac { \alpha } { \beta + 1 } + \frac { \beta } { \alpha + 1 }$$

Evaluate

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \beta +1 and \alpha +1 is \left(\alpha +1\right)\left(\beta +1\right). Multiply \frac{\alpha }{\beta +1} times \frac{\alpha +1}{\alpha +1}. Multiply \frac{\beta }{\alpha +1} times \frac{\beta +1}{\beta +1}.

$$\frac{\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}+\frac{\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Since \frac{\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} and \frac{\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\alpha \left(\alpha +1\right)+\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Do the multiplications in \alpha \left(\alpha +1\right)+\beta \left(\beta +1\right).

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Expand \left(\alpha +1\right)\left(\beta +1\right).

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\alpha \beta +\alpha +\beta +1}$$

Differentiate w.r.t. α

$$\frac{\alpha ^{2}+2\alpha -\beta ^{2}-\beta +1}{\left(\beta +1\right)\left(\alpha +1\right)^{2}}$$