Solve x/x-2div(2+x-4/2-x)

$$\frac { x } { x - 2 } \div ( 2 + x - \frac { 4 } { 2 - x } )$$

Evaluate

$$\frac{1}{x}$$

Short Solution Steps

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

$$\frac{\frac{x}{x-2}}{\frac{\left(2+x\right)\left(2-x\right)}{2-x}-\frac{4}{2-x}}$$

Since \frac{\left(2+x\right)\left(2-x\right)}{2-x} and \frac{4}{2-x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x}{x-2}}{\frac{\left(2+x\right)\left(2-x\right)-4}{2-x}}$$

Do the multiplications in \left(2+x\right)\left(2-x\right)-4.

$$\frac{\frac{x}{x-2}}{\frac{4-2x+2x-x^{2}-4}{2-x}}$$

Combine like terms in 4-2x+2x-x^{2}-4.

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

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

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

Extract the negative sign in 2-x.

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

Cancel out -x\left(x-2\right) in both numerator and denominator.

$$\frac{1}{x}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x}{x-2}}{\frac{\left(2+x\right)\left(2-x\right)}{2-x}-\frac{4}{2-x}})$$

Since \frac{\left(2+x\right)\left(2-x\right)}{2-x} and \frac{4}{2-x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x}{x-2}}{\frac{\left(2+x\right)\left(2-x\right)-4}{2-x}})$$

Do the multiplications in \left(2+x\right)\left(2-x\right)-4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x}{x-2}}{\frac{4-2x+2x-x^{2}-4}{2-x}})$$

Combine like terms in 4-2x+2x-x^{2}-4.

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

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

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

Extract the negative sign in 2-x.

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

Cancel out -x\left(x-2\right) in both numerator and denominator.

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

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

$$-x^{-1-1}$$

Subtract 1 from -1.

$$-x^{-2}$$