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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

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

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

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

Combine like terms in x-1+1.

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

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

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

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

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

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