Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = \frac{e^{y} - e^{- y}}{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e^{y} - e^{- y}}{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{e^{y} - e^{- y}}{2}$ .

$2 \frac{e^{y} - e^{- y}}{2} \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{2 (e^{y} - e^{- y})}{2} \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

$\frac{2 (e^{y} - e^{- y})}{2} \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 2.2.2

Divide $e^{y} - e^{- y}$ by $1$ .

$(e^{y} - e^{- y}) \frac{d}{d y} [\frac{e^{y} - e^{- y}}{2}]$

Step 2.3

Since $\frac{1}{2}$ is constant with respect to $y$ , the derivative of $\frac{e^{y} - e^{- y}}{2}$ with respect to $y$ is $\frac{1}{2} \frac{d}{d y} [e^{y} - e^{- y}]$ .

$(e^{y} - e^{- y}) (\frac{1}{2} \frac{d}{d y} [e^{y} - e^{- y}])$

Step 2.4

By the Sum Rule, the derivative of $e^{y} - e^{- y}$ with respect to $y$ is $\frac{d}{d y} [e^{y}] + \frac{d}{d y} [- e^{- y}]$ .

$(e^{y} - e^{- y}) \frac{1}{2} (\frac{d}{d y} [e^{y}] + \frac{d}{d y} [- e^{- y}])$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d y} [a^{y}]$ is $a^{y} \ln (a)$ where $a$ = $e$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} + \frac{d}{d y} [- e^{- y}])$

Step 4

Since $- 1$ is constant with respect to $y$ , the derivative of $- e^{- y}$ with respect to $y$ is $- \frac{d}{d y} [e^{- y}]$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} - \frac{d}{d y} [e^{- y}])$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = - y$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $- y$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} - (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d y} [- y]))$

Step 5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} - (e^{u_{2}} \frac{d}{d y} [- y]))$

Step 5.3

Replace all occurrences of $u_{2}$ with $- y$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} - (e^{- y} \frac{d}{d y} [- y]))$

Step 6

Differentiate.

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Step 6.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} - e^{- y} (- \frac{d}{d y} [y]))$

Step 6.2

Multiply.

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Step 6.2.1

Multiply $- 1$ by $- 1$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} + 1 e^{- y} \frac{d}{d y} [y])$

Step 6.2.2

Multiply $e^{- y}$ by $1$ .

$(e^{y} - e^{- y}) \frac{1}{2} (e^{y} + e^{- y} \frac{d}{d y} [y])$

Step 6.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

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

Step 6.4

Multiply $e^{- y}$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Combine terms.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

Reorder the factors of $(\frac{e^{y}}{2} - \frac{e^{- y}}{2}) (e^{y} + e^{- y})$ .

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

Step 7.4

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

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Step 7.4.1

Apply the distributive property.

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

Step 7.4.2

Apply the distributive property.

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

Step 7.4.3

Apply the distributive property.

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

Step 7.5

Simplify and combine like terms.

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Step 7.5.1

Simplify each term.

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Step 7.5.1.1

Multiply $e^{y} \frac{e^{y}}{2}$ .

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Step 7.5.1.1.1

Combine $e^{y}$ and $\frac{e^{y}}{2}$ .

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

Step 7.5.1.1.2

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

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Step 7.5.1.1.2.1

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

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

Step 7.5.1.1.2.2

Add $y$ and $y$ .

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

Step 7.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 7.5.1.3

Multiply $- e^{y} \frac{e^{- y}}{2}$ .

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Step 7.5.1.3.1

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

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

Step 7.5.1.3.2

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

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Step 7.5.1.3.2.1

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

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

Step 7.5.1.3.2.2

Add $- y$ and $y$ .

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

Step 7.5.1.3.3

Simplify $e^{0}$ .

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

Step 7.5.1.4

Multiply $e^{- y} \frac{e^{y}}{2}$ .

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Step 7.5.1.4.1

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

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

Step 7.5.1.4.2

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

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Step 7.5.1.4.2.1

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

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

Step 7.5.1.4.2.2

Add $- y$ and $y$ .

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

Step 7.5.1.4.3

Simplify $e^{0}$ .

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

Step 7.5.1.5

Rewrite using the commutative property of multiplication.

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

Step 7.5.1.6

Multiply $- e^{- y} \frac{e^{- y}}{2}$ .

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Step 7.5.1.6.1

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

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

Step 7.5.1.6.2

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

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Step 7.5.1.6.2.1

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

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

Step 7.5.1.6.2.2

Subtract $y$ from $- y$ .

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

Step 7.5.2

Add $- \frac{1}{2}$ and $\frac{1}{2}$ .

$\frac{e^{2 y}}{2} + 0 - \frac{e^{- 2 y}}{2}$

Step 7.5.3

Add $\frac{e^{2 y}}{2}$ and $0$ .

$\frac{e^{2 y}}{2} - \frac{e^{- 2 y}}{2}$