Calculus Examples

$\frac{e^{8 u} - e^{- 8 u}}{e^{8 u} + e^{- 8 u}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = e^{8 u} - e^{- 8 u}$ and $g (u) = e^{8 u} + e^{- 8 u}$ .

$\frac{(e^{8 u} + e^{- 8 u}) \frac{d}{d u} [e^{8 u} - e^{- 8 u}] - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 2

By the Sum Rule, the derivative of $e^{8 u} - e^{- 8 u}$ with respect to $u$ is $\frac{d}{d u} [e^{8 u}] + \frac{d}{d u} [- e^{- 8 u}]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (\frac{d}{d u} [e^{8 u}] + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = e^{u}$ and $g (u) = 8 u$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d u} [8 u] + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{(e^{8 u} + e^{- 8 u}) (e^{u_{1}} \frac{d}{d u} [8 u] + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 3.3

Replace all occurrences of $u_{1}$ with $8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (e^{8 u} \frac{d}{d u} [8 u] + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 4

Differentiate.

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Step 4.1

Since $8$ is constant with respect to $u$ , the derivative of $8 u$ with respect to $u$ is $8 \frac{d}{d u} [u]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (e^{8 u} (8 \frac{d}{d u} [u]) + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 4.2

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

$\frac{(e^{8 u} + e^{- 8 u}) (e^{8 u} (8 \cdot 1) + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $8$ by $1$ .

$\frac{(e^{8 u} + e^{- 8 u}) (e^{8 u} \cdot 8 + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 4.3.2

Move $8$ to the left of $e^{8 u}$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 \cdot e^{8 u} + \frac{d}{d u} [- e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 4.4

Since $- 1$ is constant with respect to $u$ , the derivative of $- e^{- 8 u}$ with respect to $u$ is $- \frac{d}{d u} [e^{- 8 u}]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} - \frac{d}{d u} [e^{- 8 u}]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = e^{u}$ and $g (u) = - 8 u$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $- 8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} - (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d u} [- 8 u])) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

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 $a$ = $e$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} - (e^{u_{2}} \frac{d}{d u} [- 8 u])) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 5.3

Replace all occurrences of $u_{2}$ with $- 8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} - (e^{- 8 u} \frac{d}{d u} [- 8 u])) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 6

Differentiate.

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Step 6.1

Since $- 8$ is constant with respect to $u$ , the derivative of $- 8 u$ with respect to $u$ is $- 8 \frac{d}{d u} [u]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} - e^{- 8 u} (- 8 \frac{d}{d u} [u])) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 6.2

Multiply $- 8$ by $- 1$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u} \frac{d}{d u} [u]) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 6.3

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

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u} \cdot 1) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 6.4

Multiply $8$ by $1$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) \frac{d}{d u} [e^{8 u} + e^{- 8 u}]}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 6.5

By the Sum Rule, the derivative of $e^{8 u} + e^{- 8 u}$ with respect to $u$ is $\frac{d}{d u} [e^{8 u}] + \frac{d}{d u} [e^{- 8 u}]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (\frac{d}{d u} [e^{8 u}] + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = e^{u}$ and $g (u) = 8 u$ .

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Step 7.1

To apply the Chain Rule, set $u_{3}$ as $8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d u} [8 u] + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (e^{u_{3}} \frac{d}{d u} [8 u] + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 7.3

Replace all occurrences of $u_{3}$ with $8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (e^{8 u} \frac{d}{d u} [8 u] + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 8

Differentiate.

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Step 8.1

Since $8$ is constant with respect to $u$ , the derivative of $8 u$ with respect to $u$ is $8 \frac{d}{d u} [u]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (e^{8 u} (8 \frac{d}{d u} [u]) + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 8.2

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

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (e^{8 u} (8 \cdot 1) + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $8$ by $1$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (e^{8 u} \cdot 8 + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 8.3.2

Move $8$ to the left of $e^{8 u}$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 \cdot e^{8 u} + \frac{d}{d u} [e^{- 8 u}])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 9

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = e^{u}$ and $g (u) = - 8 u$ .

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Step 9.1

To apply the Chain Rule, set $u_{4}$ as $- 8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + \frac{d}{d u_{4}} [e^{u_{4}}] \frac{d}{d u} [- 8 u])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 9.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{4}} [a^{u_{4}}]$ is $a^{u_{4}} \ln (a)$ where $a$ = $e$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + e^{u_{4}} \frac{d}{d u} [- 8 u])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 9.3

Replace all occurrences of $u_{4}$ with $- 8 u$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + e^{- 8 u} \frac{d}{d u} [- 8 u])}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 10

Differentiate.

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Step 10.1

Since $- 8$ is constant with respect to $u$ , the derivative of $- 8 u$ with respect to $u$ is $- 8 \frac{d}{d u} [u]$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + e^{- 8 u} (- 8 \frac{d}{d u} [u]))}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 10.2

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

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + e^{- 8 u} (- 8 \cdot 1))}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $- 8$ by $1$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} + e^{- 8 u} \cdot - 8)}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 10.3.2

Move $- 8$ to the left of $e^{- 8 u}$ .

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) - (e^{8 u} - e^{- 8 u}) (8 e^{8 u} - 8 \cdot e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Expand $(e^{8 u} + e^{- 8 u}) (8 e^{8 u} + 8 e^{- 8 u})$ using the FOIL Method.

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Step 11.2.1.1.1

Apply the distributive property.

$\frac{e^{8 u} (8 e^{8 u} + 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u} + 8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.1.2

Apply the distributive property.

$\frac{e^{8 u} (8 e^{8 u}) + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u} + 8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.1.3

Apply the distributive property.

$\frac{e^{8 u} (8 e^{8 u}) + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2

Simplify and combine like terms.

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Step 11.2.1.2.1

Simplify each term.

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Step 11.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{8 e^{8 u} e^{8 u} + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.2

Multiply $e^{8 u}$ by $e^{8 u}$ by adding the exponents.

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Step 11.2.1.2.1.2.1

Move $e^{8 u}$ .

$\frac{8 (e^{8 u} e^{8 u}) + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.2.2

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

$\frac{8 e^{8 u + 8 u} + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.2.3

Add $8 u$ and $8 u$ .

$\frac{8 e^{16 u} + e^{8 u} (8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 8 e^{8 u} e^{- 8 u} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.4

Multiply $e^{8 u}$ by $e^{- 8 u}$ by adding the exponents.

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Step 11.2.1.2.1.4.1

Move $e^{- 8 u}$ .

$\frac{8 e^{16 u} + 8 (e^{- 8 u} e^{8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.4.2

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

$\frac{8 e^{16 u} + 8 e^{- 8 u + 8 u} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.4.3

Add $- 8 u$ and $8 u$ .

$\frac{8 e^{16 u} + 8 e^{0} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.5

Simplify $8 e^{0}$ .

$\frac{8 e^{16 u} + 8 + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.6

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 8 + 8 e^{- 8 u} e^{8 u} + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.7

Multiply $e^{- 8 u}$ by $e^{8 u}$ by adding the exponents.

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Step 11.2.1.2.1.7.1

Move $e^{8 u}$ .

$\frac{8 e^{16 u} + 8 + 8 (e^{8 u} e^{- 8 u}) + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.7.2

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

$\frac{8 e^{16 u} + 8 + 8 e^{8 u - 8 u} + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.7.3

Subtract $8 u$ from $8 u$ .

$\frac{8 e^{16 u} + 8 + 8 e^{0} + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.8

Simplify $8 e^{0}$ .

$\frac{8 e^{16 u} + 8 + 8 + e^{- 8 u} (8 e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.9

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 8 + 8 + 8 e^{- 8 u} e^{- 8 u} + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.10

Multiply $e^{- 8 u}$ by $e^{- 8 u}$ by adding the exponents.

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Step 11.2.1.2.1.10.1

Move $e^{- 8 u}$ .

$\frac{8 e^{16 u} + 8 + 8 + 8 (e^{- 8 u} e^{- 8 u}) + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.10.2

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

$\frac{8 e^{16 u} + 8 + 8 + 8 e^{- 8 u - 8 u} + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.1.10.3

Subtract $8 u$ from $- 8 u$ .

$\frac{8 e^{16 u} + 8 + 8 + 8 e^{- 16 u} + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.2.2

Add $8$ and $8$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} + (- e^{8 u} - - e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.3

Multiply $- - e^{- 8 u}$ .

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Step 11.2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} + (- e^{8 u} + 1 e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.3.2

Multiply $e^{- 8 u}$ by $1$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} + (- e^{8 u} + e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.4

Expand $(- e^{8 u} + e^{- 8 u}) (8 e^{8 u} - 8 e^{- 8 u})$ using the FOIL Method.

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Step 11.2.1.4.1

Apply the distributive property.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - e^{8 u} (8 e^{8 u} - 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.4.2

Apply the distributive property.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - e^{8 u} (8 e^{8 u}) - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u} - 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.4.3

Apply the distributive property.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - e^{8 u} (8 e^{8 u}) - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5

Simplify and combine like terms.

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Step 11.2.1.5.1

Simplify each term.

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Step 11.2.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 1 \cdot 8 e^{8 u} e^{8 u} - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.2

Multiply $e^{8 u}$ by $e^{8 u}$ by adding the exponents.

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Step 11.2.1.5.1.2.1

Move $e^{8 u}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 1 \cdot 8 (e^{8 u} e^{8 u}) - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.2.2

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

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 1 \cdot 8 e^{8 u + 8 u} - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.2.3

Add $8 u$ and $8 u$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 1 \cdot 8 e^{16 u} - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.3

Multiply $- 1$ by $8$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - e^{8 u} (- 8 e^{- 8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.4

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - 1 \cdot - 8 e^{8 u} e^{- 8 u} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.5

Multiply $e^{8 u}$ by $e^{- 8 u}$ by adding the exponents.

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Step 11.2.1.5.1.5.1

Move $e^{- 8 u}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - 1 \cdot - 8 (e^{- 8 u} e^{8 u}) + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.5.2

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

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - 1 \cdot - 8 e^{- 8 u + 8 u} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.5.3

Add $- 8 u$ and $8 u$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - 1 \cdot - 8 e^{0} + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.6

Simplify $- 1 \cdot - 8 e^{0}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} - 1 \cdot - 8 + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.7

Multiply $- 1$ by $- 8$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + e^{- 8 u} (8 e^{8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.8

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 e^{- 8 u} e^{8 u} + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.9

Multiply $e^{- 8 u}$ by $e^{8 u}$ by adding the exponents.

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Step 11.2.1.5.1.9.1

Move $e^{8 u}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 (e^{8 u} e^{- 8 u}) + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.9.2

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

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 e^{8 u - 8 u} + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.9.3

Subtract $8 u$ from $8 u$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 e^{0} + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.10

Simplify $8 e^{0}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 + e^{- 8 u} (- 8 e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.11

Rewrite using the commutative property of multiplication.

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 - 8 e^{- 8 u} e^{- 8 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.12

Multiply $e^{- 8 u}$ by $e^{- 8 u}$ by adding the exponents.

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Step 11.2.1.5.1.12.1

Move $e^{- 8 u}$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 - 8 (e^{- 8 u} e^{- 8 u})}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.12.2

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

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 - 8 e^{- 8 u - 8 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.1.12.3

Subtract $8 u$ from $- 8 u$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 8 + 8 - 8 e^{- 16 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.1.5.2

Add $8$ and $8$ .

$\frac{8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 16 - 8 e^{- 16 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.2

Combine the opposite terms in $8 e^{16 u} + 16 + 8 e^{- 16 u} - 8 e^{16 u} + 16 - 8 e^{- 16 u}$ .

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Step 11.2.2.1

Subtract $8 e^{16 u}$ from $8 e^{16 u}$ .

$\frac{16 + 8 e^{- 16 u} + 0 + 16 - 8 e^{- 16 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.2.2

Add $16 + 8 e^{- 16 u}$ and $0$ .

$\frac{16 + 8 e^{- 16 u} + 16 - 8 e^{- 16 u}}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.2.3

Subtract $8 e^{- 16 u}$ from $8 e^{- 16 u}$ .

$\frac{16 + 0 + 16}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.2.4

Add $16$ and $0$ .

$\frac{16 + 16}{{(e^{8 u} + e^{- 8 u})}^{2}}$

Step 11.2.3

Add $16$ and $16$ .

$\frac{32}{{(e^{8 u} + e^{- 8 u})}^{2}}$