Lutasin ang alpha/beta+1+beta/alpha+1

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

I-evaluate

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

Maiikling Hakbang sa Solution

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng \beta +1 at \alpha +1 ay \left(\alpha +1\right)\left(\beta +1\right). I-multiply ang \frac{\alpha }{\beta +1} times \frac{\alpha +1}{\alpha +1}. I-multiply ang \frac{\beta }{\alpha +1} times \frac{\beta +1}{\beta +1}.

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

Dahil may parehong denominator ang \frac{\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} at \frac{\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

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

Gawin ang mga pag-multiply sa \alpha \left(\alpha +1\right)+\beta \left(\beta +1\right).

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

Palawakin ang \left(\alpha +1\right)\left(\beta +1\right).

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

I-differentiate ang w.r.t. α

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