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

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

Evaluate

$$-\frac{3}{x^{2}-1}$$

Short Solution Steps

Factor x^{2}-1. Factor x^{2}+x.

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

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

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

Since \frac{x}{x\left(x-1\right)\left(x+1\right)} and \frac{2\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x-2\left(x-1\right).

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

Combine like terms in x-2x+2.

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

Factor x^{2}-x.

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

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

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

Since \frac{-x+2}{x\left(x-1\right)\left(x+1\right)} and \frac{2\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -x+2-2\left(x+1\right).

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

Combine like terms in -x+2-2x-2.

$$\frac{-3x}{x\left(x-1\right)\left(x+1\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{-3}{\left(x-1\right)\left(x+1\right)}$$

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

$$\frac{-3}{x^{2}-1}$$

Differentiate w.r.t. x

$$\frac{6x}{\left(x^{2}-1\right)^{2}}$$