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

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

Evaluate

$$2-x$$

Short Solution Steps

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

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

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

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

Combine like terms in 1-x+1.

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

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

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

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

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

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

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

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

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

Combine like terms in 2-x-1+x.

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

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

$$2-x$$

Differentiate w.r.t. x

$$-1$$

Steps Using Derivative Rule for Sum

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

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

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

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

Combine like terms in 1-x+1.

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

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

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

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

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

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

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

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

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

Combine like terms in 2-x-1+x.

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

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

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. 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.

$$-1$$