Calculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x (x + 1)$ .

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Step 1.1

Multiply $\frac{\frac{1}{x} - \frac{1}{x + 1}}{\frac{1}{x} + \frac{1}{x + 1}}$ by $\frac{x (x + 1)}{x (x + 1)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x + 1$ out of $x (x + 1)$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 1$ .

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Step 3.4.1

Factor $x + 1$ out of $x (x + 1)$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Subtract $x$ from $x$ .

$\frac{0 + 1}{x + 1 + x}$

Step 4.4

Add $0$ and $1$ .

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

Step 5

Add $x$ and $x$ .

$\frac{1}{2 x + 1}$