Solve (x^2/x-2+4/2-x)divx+2/x

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

Evaluate

$$x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and 2-x is x-2. Multiply \frac{4}{2-x} times \frac{-1}{-1}.

$$\frac{\frac{x^{2}}{x-2}+\frac{4\left(-1\right)}{x-2}}{\frac{x+2}{x}}$$

Since \frac{x^{2}}{x-2} and \frac{4\left(-1\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}+4\left(-1\right)}{x-2}}{\frac{x+2}{x}}$$

Do the multiplications in x^{2}+4\left(-1\right).

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

Factor the expressions that are not already factored in \frac{x^{2}-4}{x-2}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Cancel out x+2 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. Least common multiple of x-2 and 2-x is x-2. Multiply \frac{4}{2-x} times \frac{-1}{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x^{2}}{x-2}+\frac{4\left(-1\right)}{x-2}}{\frac{x+2}{x}})$$

Since \frac{x^{2}}{x-2} and \frac{4\left(-1\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x^{2}+4\left(-1\right)}{x-2}}{\frac{x+2}{x}})$$

Do the multiplications in x^{2}+4\left(-1\right).

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

Factor the expressions that are not already factored in \frac{x^{2}-4}{x-2}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Cancel out x+2 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.

$$1$$