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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-1-2x.

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

Extract the negative sign in -x-1.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

Differentiate w.r.t. x

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