Solve 2/x-1+2/frac{x+1{x-1}-1}

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

Evaluate

$$x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in 2+2x-2.

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

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

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

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

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

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

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

Combine like terms in x+1-x+1.

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

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

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

Cancel out 2\left(x-1\right) 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 2 times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in 2+2x-2.

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

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

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

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

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

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

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

Combine like terms in x+1-x+1.

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

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

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

Cancel out 2\left(x-1\right) 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.

$$1$$