Calculus Examples

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

Step 1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} - x \cdot \frac{x}{x}$

Step 2

Simplify terms.

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Step 2.1

Combine $- x$ and $\frac{x}{x}$ .

$\frac{1}{x} + \frac{- x \cdot x}{x}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{1 - x \cdot x}{x}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{(1 + x) (1 - x)}{x}$