Solve (x-2/x+2+8x/x^2-4)divx+2/x^2-2x

$$( \frac{ x-2 }{ x+2 } + \frac{ 8x }{ { x }^{ 2 } -4 } ) \div \frac{ x+2 }{ { x }^{ 2 } -2x }$$

Evaluate

$$x$$

Short Solution Steps

Factor x^{2}-4.

$$\frac{\frac{x-2}{x+2}+\frac{8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}}$$

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

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}}$$

Since \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{8x}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)+8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}}$$

Do the multiplications in \left(x-2\right)\left(x-2\right)+8x.

$$\frac{\frac{x^{2}-2x-2x+4+8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}}$$

Combine like terms in x^{2}-2x-2x+4+8x.

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

Factor the expressions that are not already factored in \frac{x^{2}+4x+4}{\left(x-2\right)\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

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

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

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

Cancel out x+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Factor x^{2}-4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x-2}{x+2}+\frac{8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}})$$

Since \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{8x}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{\left(x-2\right)\left(x-2\right)+8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}})$$

Do the multiplications in \left(x-2\right)\left(x-2\right)+8x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x^{2}-2x-2x+4+8x}{\left(x-2\right)\left(x+2\right)}}{\frac{x+2}{x^{2}-2x}})$$

Combine like terms in x^{2}-2x-2x+4+8x.

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

Factor the expressions that are not already factored in \frac{x^{2}+4x+4}{\left(x-2\right)\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

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

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

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

Cancel out x+2 in both numerator and denominator.

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

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

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

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.

$$1$$