Solve 1/x^2+x+1/x^2-x

$$\frac { 1 } { x ^ { 2 } + x } + \frac { 1 } { x ^ { 2 } - x }$$

Evaluate

$$\frac{2}{x^{2}-1}$$

Short Solution Steps

Factor x^{2}+x. Factor x^{2}-x.

$$\frac{1}{x\left(x+1\right)}+\frac{1}{x\left(x-1\right)}$$

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

$$\frac{x-1}{x\left(x-1\right)\left(x+1\right)}+\frac{x+1}{x\left(x-1\right)\left(x+1\right)}$$

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

$$\frac{x-1+x+1}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in x-1+x+1.

$$\frac{2x}{x\left(x-1\right)\left(x+1\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{2}{\left(x-1\right)\left(x+1\right)}$$

Expand \left(x-1\right)\left(x+1\right).

$$\frac{2}{x^{2}-1}$$

Differentiate w.r.t. x

$$-\frac{4x}{\left(x^{2}-1\right)^{2}}$$