Solve 1/x-1/x^2-x-1/x-5

$$\frac{ 1 }{ x } - \frac{ 1 }{ { x }^{ 2 } -x } - \frac{ 1 }{ x-5 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-x.

$$\frac{1}{x}-\frac{1}{x\left(x-1\right)}-\frac{1}{x-5}$$

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

$$\frac{x-1}{x\left(x-1\right)}-\frac{1}{x\left(x-1\right)}-\frac{1}{x-5}$$

Since \frac{x-1}{x\left(x-1\right)} and \frac{1}{x\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-1-1}{x\left(x-1\right)}-\frac{1}{x-5}$$

Combine like terms in x-1-1.

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

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

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

Since \frac{\left(x-2\right)\left(x-5\right)}{x\left(x-5\right)\left(x-1\right)} and \frac{x\left(x-1\right)}{x\left(x-5\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x-5\right)-x\left(x-1\right).

$$\frac{x^{2}-5x-2x+10-x^{2}+x}{x\left(x-5\right)\left(x-1\right)}$$

Combine like terms in x^{2}-5x-2x+10-x^{2}+x.

$$\frac{-6x+10}{x\left(x-5\right)\left(x-1\right)}$$

Expand x\left(x-5\right)\left(x-1\right).

$$\frac{-6x+10}{x^{3}-6x^{2}+5x}$$

Differentiate w.r.t. x

$$\frac{2\left(6x^{3}-33x^{2}+60x-25\right)}{\left(x\left(x-5\right)\left(x-1\right)\right)^{2}}$$