Calculus Examples

$\frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$

Step 1

Multiply the numerator and denominator of the fraction by $x \cdot 3$ .

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Step 1.1

Multiply $\frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$ by $\frac{x \cdot 3}{x \cdot 3}$ .

$\frac{x \cdot 3}{x \cdot 3} \cdot \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$

Step 1.2

Combine.

$\frac{x \cdot 3 (\frac{1}{x} - \frac{1}{3})}{x \cdot 3 (x - 3)}$

Step 2

Apply the distributive property.

$\frac{x \cdot 3 \frac{1}{x} + x \cdot 3 (- \frac{1}{3})}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x \cdot 3$ .

$\frac{x (3) \frac{1}{x} + x \cdot 3 (- \frac{1}{3})}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.1.2

Cancel the common factor.

$\frac{x \cdot 3 \frac{1}{x} + x \cdot 3 (- \frac{1}{3})}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.1.3

Rewrite the expression.

$\frac{3 + x \cdot 3 (- \frac{1}{3})}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{3 + x \cdot 3 \frac{- 1}{3}}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.2.2

Factor $3$ out of $x \cdot 3$ .

$\frac{3 + 3 \cdot x \frac{- 1}{3}}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.2.3

Cancel the common factor.

$\frac{3 + 3 \cdot x \frac{- 1}{3}}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 3.2.4

Rewrite the expression.

$\frac{3 + x \cdot - 1}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 4

Simplify the numerator.

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Step 4.1

Move $- 1$ to the left of $x$ .

$\frac{3 - 1 \cdot x}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 4.2

Rewrite $- 1 x$ as $- x$ .

$\frac{3 - x}{x \cdot 3 x + x \cdot 3 \cdot - 3}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.1

Move $x$ .

$\frac{3 - x}{x \cdot x \cdot 3 + x \cdot 3 \cdot - 3}$

Step 5.1.2

Multiply $x$ by $x$ .

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

Step 5.2

Move $3$ to the left of $x^{2}$ .

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

Step 5.3

Move $3$ to the left of $x$ .

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

Step 5.4

Multiply $- 3$ by $3$ .

$\frac{3 - x}{3 x^{2} - 9 x}$

Step 5.5

Factor $3 x$ out of $3 x^{2} - 9 x$ .

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Step 5.5.1

Factor $3 x$ out of $3 x^{2}$ .

$\frac{3 - x}{3 x (x) - 9 x}$

Step 5.5.2

Factor $3 x$ out of $- 9 x$ .

$\frac{3 - x}{3 x (x) + 3 x (- 3)}$

Step 5.5.3

Factor $3 x$ out of $3 x (x) + 3 x (- 3)$ .

$\frac{3 - x}{3 x (x - 3)}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 6.1.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- 3) - x}{3 x (x - 3)}$

Step 6.1.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 3) - (x)}{3 x (x - 3)}$

Step 6.1.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{- 1 (- 3 + x)}{3 x (x - 3)}$

Step 6.1.4

Reorder terms.

$\frac{- 1 (x - 3)}{3 x (x - 3)}$

Step 6.1.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{3 x (x - 3)}$

Step 6.1.6

Rewrite the expression.

$\frac{- 1}{3 x}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{3 x}$