Solve 1/x^2-2x-1/x^2-4x+4

$$\frac{ 1 }{ { x }^{ 2 } -2x } - \frac{ 1 }{ { x }^{ 2 } -4x+4 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-2x. Factor x^{2}-4x+4.

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

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

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

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

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

Combine like terms in x-2-x.

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

Expand x\left(x-2\right)^{2}.

$$\frac{-2}{x^{3}-4x^{2}+4x}$$

Differentiate w.r.t. x

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