Solve x/x^2-1-6/x^2-2x+1

$$\frac { x } { x ^ { 2 } - 1 } - \frac { 6 } { x ^ { 2 } - 2 x + 1 }$$

Evaluate

$$\frac{x^{2}-7x-6}{\left(x+1\right)\left(x-1\right)^{2}}$$

Short Solution Steps

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

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

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 \left(x-1\right)^{2} is \left(x+1\right)\left(x-1\right)^{2}. Multiply \frac{x}{\left(x-1\right)\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{6}{\left(x-1\right)^{2}} times \frac{x+1}{x+1}.

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

Since \frac{x\left(x-1\right)}{\left(x+1\right)\left(x-1\right)^{2}} and \frac{6\left(x+1\right)}{\left(x+1\right)\left(x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x-1\right)-6\left(x+1\right).

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

Combine like terms in x^{2}-x-6x-6.

$$\frac{x^{2}-7x-6}{\left(x+1\right)\left(x-1\right)^{2}}$$

Expand \left(x+1\right)\left(x-1\right)^{2}.

$$\frac{x^{2}-7x-6}{x^{3}-x^{2}-x+1}$$

Differentiate w.r.t. x

$$\frac{13+23x+13x^{2}-x^{3}}{\left(x+1\right)^{2}\left(x-1\right)^{3}}$$