Calculus Examples

$\frac{e - e^{- 1}}{e + e^{- 1}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{e - \frac{1}{e}}{e + e^{- 1}}$

Step 1.2

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

$\frac{\frac{e e}{e} - \frac{1}{e}}{e + e^{- 1}}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\frac{e e - 1}{e}}{e + e^{- 1}}$

Step 1.4

Rewrite $\frac{e e - 1}{e}$ in a factored form.

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Step 1.4.1

Multiply $e e$ .

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Step 1.4.1.1

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{1} e - 1}{e}}{e + e^{- 1}}$

Step 1.4.1.2

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{1} e^{1} - 1}{e}}{e + e^{- 1}}$

Step 1.4.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{e^{1 + 1} - 1}{e}}{e + e^{- 1}}$

Step 1.4.1.4

Add $1$ and $1$ .

$\frac{\frac{e^{2} - 1}{e}}{e + e^{- 1}}$

Step 1.4.2

Rewrite $e^{2} - 1$ in a factored form.

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Step 1.4.2.1

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

$\frac{\frac{e^{2} - 1^{2}}{e}}{e + e^{- 1}}$

Step 1.4.2.2

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

$\frac{\frac{(e + 1) (e - 1)}{e}}{e + e^{- 1}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Multiply $e e$ .

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Step 2.4.1

Raise $e$ to the power of $1$ .

$\frac{\frac{(e + 1) (e - 1)}{e}}{\frac{e^{1} e + 1}{e}}$

Step 2.4.2

Raise $e$ to the power of $1$ .

$\frac{\frac{(e + 1) (e - 1)}{e}}{\frac{e^{1} e^{1} + 1}{e}}$

Step 2.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(e + 1) (e - 1)}{e}}{\frac{e^{1 + 1} + 1}{e}}$

Step 2.4.4

Add $1$ and $1$ .

$\frac{\frac{(e + 1) (e - 1)}{e}}{\frac{e^{2} + 1}{e}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{(e + 1) (e - 1)}{e} \cdot \frac{e}{e^{2} + 1}$

Step 4

Cancel the common factor of $e$ .

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Step 4.1

Cancel the common factor.

$\frac{(e + 1) (e - 1)}{e} \cdot \frac{e}{e^{2} + 1}$

Step 4.2

Rewrite the expression.

$(e + 1) (e - 1) \frac{1}{e^{2} + 1}$

Step 5

Expand $(e + 1) (e - 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$(e (e - 1) + 1 (e - 1)) \frac{1}{e^{2} + 1}$

Step 5.2

Apply the distributive property.

$(e e + e \cdot - 1 + 1 (e - 1)) \frac{1}{e^{2} + 1}$

Step 5.3

Apply the distributive property.

$(e e + e \cdot - 1 + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $e e$ .

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Step 6.1.1.1

Raise $e$ to the power of $1$ .

$(e^{1} e + e \cdot - 1 + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.1.2

Raise $e$ to the power of $1$ .

$(e^{1} e^{1} + e \cdot - 1 + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(e^{1 + 1} + e \cdot - 1 + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.1.4

Add $1$ and $1$ .

$(e^{2} + e \cdot - 1 + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.2

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

$(e^{2} - 1 \cdot e + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.3

Rewrite $- 1 e$ as $- e$ .

$(e^{2} - e + 1 e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.4

Multiply $e$ by $1$ .

$(e^{2} - e + e + 1 \cdot - 1) \frac{1}{e^{2} + 1}$

Step 6.1.5

Multiply $- 1$ by $1$ .

$(e^{2} - e + e - 1) \frac{1}{e^{2} + 1}$

Step 6.2

Add $- e$ and $e$ .

$(e^{2} + 0 - 1) \frac{1}{e^{2} + 1}$

Step 6.3

Add $e^{2}$ and $0$ .

$(e^{2} - 1) \frac{1}{e^{2} + 1}$

Step 7

Simplify terms.

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Step 7.1

Apply the distributive property.

$e^{2} \frac{1}{e^{2} + 1} - 1 \frac{1}{e^{2} + 1}$

Step 7.2

Combine $e^{2}$ and $\frac{1}{e^{2} + 1}$ .

$\frac{e^{2}}{e^{2} + 1} - 1 \frac{1}{e^{2} + 1}$

Step 7.3

Simplify the expression.

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Step 7.3.1

Rewrite $- 1 \frac{1}{e^{2} + 1}$ as $- \frac{1}{e^{2} + 1}$ .

$\frac{e^{2}}{e^{2} + 1} - \frac{1}{e^{2} + 1}$

Step 7.3.2

Combine the numerators over the common denominator.

$\frac{e^{2} - 1}{e^{2} + 1}$

Step 8

Simplify the numerator.

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Step 8.1

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

$\frac{e^{2} - 1^{2}}{e^{2} + 1}$

Step 8.2

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

$\frac{(e + 1) (e - 1)}{e^{2} + 1}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{(e + 1) (e - 1)}{e^{2} + 1}$

Decimal Form:

$0.76159415 \dots$