Solve [1/x-2-2/x^2-2x]*1

$$[ \frac { 1 } { x - 2 } - \frac { 2 } { x ^ { 2 } - 2 x } ] \cdot 1$$

Evaluate

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

Short Solution Steps

Factor x^{2}-2x.

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

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

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

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

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

Cancel out x-2 in both numerator and denominator.

$$\frac{1}{x}\times 1$$

Express \frac{1}{x}\times 1 as a single fraction.

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

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Factor x^{2}-2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x-2}-\frac{2}{x\left(x-2\right)}\right)\times 1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x}{x\left(x-2\right)}-\frac{2}{x\left(x-2\right)}\right)\times 1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x-2}{x\left(x-2\right)}\times 1)$$

Cancel out x-2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x}\times 1)$$

Express \frac{1}{x}\times 1 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})$$

The derivative of ax^{n} is nax^{n-1}.

$$-x^{-1-1}$$

Subtract 1 from -1.

$$-x^{-2}$$