Solve [1/x-2/x+2+3/x+3]

$$[ \frac { 1 } { x } - \frac { 2 } { x + 2 } + \frac { 3 } { x + 3 } ]$$

Evaluate

$$\frac{2x^{2}+5x+6}{x\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

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

$$\frac{x+2}{x\left(x+2\right)}-\frac{2x}{x\left(x+2\right)}+\frac{3}{x+3}$$

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

$$\frac{x+2-2x}{x\left(x+2\right)}+\frac{3}{x+3}$$

Combine like terms in x+2-2x.

$$\frac{-x+2}{x\left(x+2\right)}+\frac{3}{x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and x+3 is x\left(x+2\right)\left(x+3\right). Multiply \frac{-x+2}{x\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{3}{x+3} times \frac{x\left(x+2\right)}{x\left(x+2\right)}.

$$\frac{\left(-x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}$$

Since \frac{\left(-x+2\right)\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)} and \frac{3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-x+2\right)\left(x+3\right)+3x\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in \left(-x+2\right)\left(x+3\right)+3x\left(x+2\right).

$$\frac{-x^{2}-3x+2x+6+3x^{2}+6x}{x\left(x+2\right)\left(x+3\right)}$$

Combine like terms in -x^{2}-3x+2x+6+3x^{2}+6x.

$$\frac{2x^{2}+5x+6}{x\left(x+2\right)\left(x+3\right)}$$

Expand x\left(x+2\right)\left(x+3\right).

$$\frac{2x^{2}+5x+6}{x^{3}+5x^{2}+6x}$$

Differentiate w.r.t. x

$$-\frac{2x^{4}+10x^{3}+31x^{2}+60x+36}{\left(x\left(x+2\right)\left(x+3\right)\right)^{2}}$$