Solve 1/x+2-1/x+3-1/x+4+1/x+5

$$\frac{ 1 }{ x+2 } - \frac{ 1 }{ x+3 } - \frac{ 1 }{ x+4 } + \frac{ 1 }{ x+5 }$$

Evaluate

$$\frac{2\left(2x+7\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+3 is \left(x+2\right)\left(x+3\right). Multiply \frac{1}{x+2} times \frac{x+3}{x+3}. Multiply \frac{1}{x+3} times \frac{x+2}{x+2}.

$$\frac{x+3}{\left(x+2\right)\left(x+3\right)}-\frac{x+2}{\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}+\frac{1}{x+5}$$

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

$$\frac{x+3-\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}+\frac{1}{x+5}$$

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

$$\frac{x+3-x-2}{\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}+\frac{1}{x+5}$$

Combine like terms in x+3-x-2.

$$\frac{1}{\left(x+2\right)\left(x+3\right)}-\frac{1}{x+4}+\frac{1}{x+5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x+3\right) and x+4 is \left(x+2\right)\left(x+3\right)\left(x+4\right). Multiply \frac{1}{\left(x+2\right)\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{1}{x+4} times \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}.

$$\frac{x+4}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}-\frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{1}{x+5}$$

Since \frac{x+4}{\left(x+2\right)\left(x+3\right)\left(x+4\right)} and \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+4-\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{1}{x+5}$$

Do the multiplications in x+4-\left(x+2\right)\left(x+3\right).

$$\frac{x+4-x^{2}-3x-2x-6}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{1}{x+5}$$

Combine like terms in x+4-x^{2}-3x-2x-6.

$$\frac{-4x-2-x^{2}}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{1}{x+5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x+3\right)\left(x+4\right) and x+5 is \left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right). Multiply \frac{-4x-2-x^{2}}{\left(x+2\right)\left(x+3\right)\left(x+4\right)} times \frac{x+5}{x+5}. Multiply \frac{1}{x+5} times \frac{\left(x+2\right)\left(x+3\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{\left(-4x-2-x^{2}\right)\left(x+5\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}+\frac{\left(x+2\right)\left(x+3\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}$$

Since \frac{\left(-4x-2-x^{2}\right)\left(x+5\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)} and \frac{\left(x+2\right)\left(x+3\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-4x-2-x^{2}\right)\left(x+5\right)+\left(x+2\right)\left(x+3\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in \left(-4x-2-x^{2}\right)\left(x+5\right)+\left(x+2\right)\left(x+3\right)\left(x+4\right).

$$\frac{-4x^{2}-20x-2x-10-x^{3}-5x^{2}+x^{3}+7x^{2}+12x+2x^{2}+14x+24}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}$$

Combine like terms in -4x^{2}-20x-2x-10-x^{3}-5x^{2}+x^{3}+7x^{2}+12x+2x^{2}+14x+24.

$$\frac{4x+14}{\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)}$$

Expand \left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right).

$$\frac{4x+14}{x^{4}+14x^{3}+71x^{2}+154x+120}$$

Differentiate w.r.t. x

$$\frac{4\left(-3x^{4}-42x^{3}-218x^{2}-497x-419\right)}{\left(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\right)^{2}}$$