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

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

Evaluate

$$x$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}-x+x-1-x^{2}+x-1.

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

Divide \frac{x-2}{x-1} by \frac{x-2}{x^{2}-x} by multiplying \frac{x-2}{x-1} by the reciprocal of \frac{x-2}{x^{2}-x}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

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

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

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

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

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

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

Combine like terms in x^{2}-x+x-1-x^{2}+x-1.

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

Divide \frac{x-2}{x-1} by \frac{x-2}{x^{2}-x} by multiplying \frac{x-2}{x-1} by the reciprocal of \frac{x-2}{x^{2}-x}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x^{2}-x}{x-1}.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The derivative of ax^{n} is nax^{n-1}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

For any term t except 0, t^{0}=1.

$$1$$