Solve 2/3x^2-3x+5/6x-2/3x^-3=

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

Evaluate

$$\frac{1}{6x}$$

Short Solution Steps

Calculate x to the power of 1 and get x.

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

Factor 3x^{2}-3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x\left(x-1\right) and 6x is 6x\left(x-1\right). Multiply \frac{2}{3x\left(x-1\right)} times \frac{2}{2}. Multiply \frac{5}{6x} times \frac{x-1}{x-1}.

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

Since \frac{2\times 2}{6x\left(x-1\right)} and \frac{5\left(x-1\right)}{6x\left(x-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in 4+5x-5.

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

Factor 3x-3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6x\left(x-1\right) and 3\left(x-1\right) is 6x\left(x-1\right). Multiply \frac{2}{3\left(x-1\right)} times \frac{2x}{2x}.

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

Since \frac{-1+5x}{6x\left(x-1\right)} and \frac{2\times 2x}{6x\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -1+5x-2\times 2x.

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

Combine like terms in -1+5x-4x.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{1}{6x}$$

Differentiate w.r.t. x

$$-\frac{1}{6x^{2}}$$