Solve 12/x^2+2x-2/x+6/x+2

$$\frac{ 12 }{ { x }^{ 2 } +2x } - \frac{ 2 }{ x } + \frac{ 6 }{ x+2 }$$

Evaluate

$$\frac{4}{x}$$

Short Solution Steps

Factor x^{2}+2x.

$$\frac{12}{x\left(x+2\right)}-\frac{2}{x}+\frac{6}{x+2}$$

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

$$\frac{12}{x\left(x+2\right)}-\frac{2\left(x+2\right)}{x\left(x+2\right)}+\frac{6}{x+2}$$

Since \frac{12}{x\left(x+2\right)} and \frac{2\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{12-2\left(x+2\right)}{x\left(x+2\right)}+\frac{6}{x+2}$$

Do the multiplications in 12-2\left(x+2\right).

$$\frac{12-2x-4}{x\left(x+2\right)}+\frac{6}{x+2}$$

Combine like terms in 12-2x-4.

$$\frac{8-2x}{x\left(x+2\right)}+\frac{6}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and x+2 is x\left(x+2\right). Multiply \frac{6}{x+2} times \frac{x}{x}.

$$\frac{8-2x}{x\left(x+2\right)}+\frac{6x}{x\left(x+2\right)}$$

Since \frac{8-2x}{x\left(x+2\right)} and \frac{6x}{x\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{8-2x+6x}{x\left(x+2\right)}$$

Combine like terms in 8-2x+6x.

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

Factor the expressions that are not already factored in \frac{8+4x}{x\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{4}{x}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Factor x^{2}+2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12}{x\left(x+2\right)}-\frac{2}{x}+\frac{6}{x+2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12}{x\left(x+2\right)}-\frac{2\left(x+2\right)}{x\left(x+2\right)}+\frac{6}{x+2})$$

Since \frac{12}{x\left(x+2\right)} and \frac{2\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12-2\left(x+2\right)}{x\left(x+2\right)}+\frac{6}{x+2})$$

Do the multiplications in 12-2\left(x+2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12-2x-4}{x\left(x+2\right)}+\frac{6}{x+2})$$

Combine like terms in 12-2x-4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8-2x}{x\left(x+2\right)}+\frac{6}{x+2})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and x+2 is x\left(x+2\right). Multiply \frac{6}{x+2} times \frac{x}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8-2x}{x\left(x+2\right)}+\frac{6x}{x\left(x+2\right)})$$

Since \frac{8-2x}{x\left(x+2\right)} and \frac{6x}{x\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8-2x+6x}{x\left(x+2\right)})$$

Combine like terms in 8-2x+6x.

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

Factor the expressions that are not already factored in \frac{8+4x}{x\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{x})$$

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

$$-4x^{-1-1}$$

Subtract 1 from -1.

$$-4x^{-2}$$