Solve 12/y^2+2y-2/y+6/y+2=

$$\frac { 12 } { y ^ { 2 } + 2 y } - \frac { 2 } { y } + \frac { 6 } { y + 2 } =$$

Evaluate

$$\frac{4}{y}$$

Short Solution Steps

Factor y^{2}+2y.

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

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

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

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

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

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

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

Combine like terms in 12-2y-4.

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

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

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

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

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

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

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

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

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

Cancel out y+2 in both numerator and denominator.

$$\frac{4}{y}$$

Differentiate w.r.t. y

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

Steps Using Definition of a Derivative

Factor y^{2}+2y.

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

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

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

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

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

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

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

Combine like terms in 12-2y-4.

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

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

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

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

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

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

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

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

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

Cancel out y+2 in both numerator and denominator.

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

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

$$-4y^{-1-1}$$

Subtract 1 from -1.

$$-4y^{-2}$$