Calculus Examples

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

Step 1

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

$(e^{x} - e^{- x}) (e^{x} - e^{- x})$

Step 2

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

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Step 2.1

Apply the distributive property.

$e^{x} (e^{x} - e^{- x}) - e^{- x} (e^{x} - e^{- x})$

Step 2.2

Apply the distributive property.

$e^{x} e^{x} + e^{x} (- e^{- x}) - e^{- x} (e^{x} - e^{- x})$

Step 2.3

Apply the distributive property.

$e^{x} e^{x} + e^{x} (- e^{- x}) - e^{- x} e^{x} - e^{- x} (- e^{- x})$

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

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

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Step 3.1.1.1

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

$e^{x + x} + e^{x} (- e^{- x}) - e^{- x} e^{x} - e^{- x} (- e^{- x})$

Step 3.1.1.2

Add $x$ and $x$ .

$e^{2 x} + e^{x} (- e^{- x}) - e^{- x} e^{x} - e^{- x} (- e^{- x})$

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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Step 3.1.3.1

Move $e^{- x}$ .

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

Step 3.1.3.2

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

$e^{2 x} - e^{- x + x} - e^{- x} e^{x} - e^{- x} (- e^{- x})$

Step 3.1.3.3

Add $- x$ and $x$ .

$e^{2 x} - e^{0} - e^{- x} e^{x} - e^{- x} (- e^{- x})$

Step 3.1.4

Simplify $- e^{0}$ .

$e^{2 x} - 1 - e^{- x} e^{x} - e^{- x} (- e^{- x})$

Step 3.1.5

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

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Step 3.1.5.1

Move $e^{x}$ .

$e^{2 x} - 1 - (e^{x} e^{- x}) - e^{- x} (- e^{- x})$

Step 3.1.5.2

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

$e^{2 x} - 1 - e^{x - x} - e^{- x} (- e^{- x})$

Step 3.1.5.3

Subtract $x$ from $x$ .

$e^{2 x} - 1 - e^{0} - e^{- x} (- e^{- x})$

Step 3.1.6

Simplify $- e^{0}$ .

$e^{2 x} - 1 - 1 - e^{- x} (- e^{- x})$

Step 3.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.1.8

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

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Step 3.1.8.1

Move $e^{- x}$ .

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

Step 3.1.8.2

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

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

Step 3.1.8.3

Subtract $x$ from $- x$ .

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

Step 3.1.9

Multiply $- 1$ by $- 1$ .

$e^{2 x} - 1 - 1 + 1 e^{- 2 x}$

Step 3.1.10

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

$e^{2 x} - 1 - 1 + e^{- 2 x}$

Step 3.2

Subtract $1$ from $- 1$ .

$e^{2 x} - 2 + e^{- 2 x}$