Solve 1/a+2-1/a-2+2/a^2-4

$$\frac { 1 } { a + 2 } - \frac { 1 } { a - 2 } + \frac { 2 } { a ^ { 2 } - 4 }$$

Evaluate

$$-\frac{2}{a^{2}-4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2 and a-2 is \left(a-2\right)\left(a+2\right). Multiply \frac{1}{a+2} times \frac{a-2}{a-2}. Multiply \frac{1}{a-2} times \frac{a+2}{a+2}.

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

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

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

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

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

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

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

Factor a^{2}-4.

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

Since \frac{-4}{\left(a-2\right)\left(a+2\right)} and \frac{2}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators. Add -4 and 2 to get -2.

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

Expand \left(a-2\right)\left(a+2\right).

$$\frac{-2}{a^{2}-4}$$

Differentiate w.r.t. a

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