Solve 2/a+beta-a/a^2-beta^2

$$\frac { 2 } { a + \beta } - \frac { a } { a ^ { 2 } - \beta ^ { 2 } }$$

Evaluate

$$\frac{a-2\beta }{a^{2}-\beta ^{2}}$$

Short Solution Steps

Factor a^{2}-\beta ^{2}.

$$\frac{2}{a+\beta }-\frac{a}{\left(a+\beta \right)\left(a-\beta \right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+\beta and \left(a+\beta \right)\left(a-\beta \right) is \left(a+\beta \right)\left(a-\beta \right). Multiply \frac{2}{a+\beta } times \frac{a-\beta }{a-\beta }.

$$\frac{2\left(a-\beta \right)}{\left(a+\beta \right)\left(a-\beta \right)}-\frac{a}{\left(a+\beta \right)\left(a-\beta \right)}$$

Since \frac{2\left(a-\beta \right)}{\left(a+\beta \right)\left(a-\beta \right)} and \frac{a}{\left(a+\beta \right)\left(a-\beta \right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(a-\beta \right)-a}{\left(a+\beta \right)\left(a-\beta \right)}$$

Do the multiplications in 2\left(a-\beta \right)-a.

$$\frac{2a-2\beta -a}{\left(a+\beta \right)\left(a-\beta \right)}$$

Combine like terms in 2a-2\beta -a.

$$\frac{a-2\beta }{\left(a+\beta \right)\left(a-\beta \right)}$$

Expand \left(a+\beta \right)\left(a-\beta \right).

$$\frac{a-2\beta }{a^{2}-\beta ^{2}}$$

Differentiate w.r.t. a

$$\frac{-a^{2}+4a\beta -\beta ^{2}}{\left(a^{2}-\beta ^{2}\right)^{2}}$$