Calculus Examples

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

Step 1

Simplify terms.

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Step 1.1

Apply the distributive property.

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

Step 1.2

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

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

Step 1.3

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

${(\frac{e^{x}}{2} - \frac{e^{- x}}{2})}^{2}$

Step 1.4

Rewrite ${(\frac{e^{x}}{2} - \frac{e^{- x}}{2})}^{2}$ as $(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$ .

$(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$

Step 2

Expand $(\frac{e^{x}}{2} - \frac{e^{- x}}{2}) (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{e^{x}}{2} (\frac{e^{x}}{2} - \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} (\frac{e^{x}}{2} - \frac{e^{- x}}{2})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine.

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

Step 3.1.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Add $x$ and $x$ .

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

Step 3.1.3

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} + \frac{e^{x}}{2} (- \frac{e^{- x}}{2}) - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{e^{2 x}}{4} - \frac{e^{x}}{2} \cdot \frac{e^{- x}}{2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.5

Multiply $- \frac{e^{x}}{2} \cdot \frac{e^{- x}}{2}$ .

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Step 3.1.5.1

Multiply $\frac{e^{- x}}{2}$ by $\frac{e^{x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{e^{- x} e^{x}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.5.2

Multiply $e^{- x}$ by $e^{x}$ by adding the exponents.

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Step 3.1.5.2.1

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

$\frac{e^{2 x}}{4} - \frac{e^{- x + x}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.5.2.2

Add $- x$ and $x$ .

$\frac{e^{2 x}}{4} - \frac{e^{0}}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.5.3

Simplify $e^{0}$ .

$\frac{e^{2 x}}{4} - \frac{1}{2 \cdot 2} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.5.4

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.6

Multiply $- \frac{e^{- x}}{2} \cdot \frac{e^{x}}{2}$ .

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Step 3.1.6.1

Multiply $\frac{e^{x}}{2}$ by $\frac{e^{- x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{x} e^{- x}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.6.2

Multiply $e^{x}$ by $e^{- x}$ by adding the exponents.

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Step 3.1.6.2.1

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

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{x - x}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.6.2.2

Subtract $x$ from $x$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{e^{0}}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.6.3

Simplify $e^{0}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{2 \cdot 2} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.6.4

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} - \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$

Step 3.1.7

Multiply $- \frac{e^{- x}}{2} (- \frac{e^{- x}}{2})$ .

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Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + 1 \frac{e^{- x}}{2} \frac{e^{- x}}{2}$

Step 3.1.7.2

Multiply $\frac{e^{- x}}{2}$ by $1$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x}}{2} \cdot \frac{e^{- x}}{2}$

Step 3.1.7.3

Multiply $\frac{e^{- x}}{2}$ by $\frac{e^{- x}}{2}$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x} e^{- x}}{2 \cdot 2}$

Step 3.1.7.4

Multiply $e^{- x}$ by $e^{- x}$ by adding the exponents.

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Step 3.1.7.4.1

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

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- x - x}}{2 \cdot 2}$

Step 3.1.7.4.2

Subtract $x$ from $- x$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- 2 x}}{2 \cdot 2}$

Step 3.1.7.5

Multiply $2$ by $2$ .

$\frac{e^{2 x}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{e^{- 2 x}}{4}$

Step 3.2

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$\frac{e^{2 x}}{4} - 2 (\frac{1}{4}) + \frac{e^{- 2 x}}{4}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$\frac{e^{2 x}}{4} + 2 (- 1) \frac{1}{4} + \frac{e^{- 2 x}}{4}$

Step 4.1.2

Factor $2$ out of $4$ .

$\frac{e^{2 x}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{e^{- 2 x}}{4}$

Step 4.1.3

Cancel the common factor.

$\frac{e^{2 x}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{e^{- 2 x}}{4}$

Step 4.1.4

Rewrite the expression.

$\frac{e^{2 x}}{4} - 1 (\frac{1}{2}) + \frac{e^{- 2 x}}{4}$

Step 4.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\frac{e^{2 x}}{4} - \frac{1}{2} + \frac{e^{- 2 x}}{4}$