Calculus Examples

$y = \cos (\frac{1 - e^{8 x}}{1 + e^{8 x}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{1 - e^{8 x}}{1 + e^{8 x}}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{1 - e^{8 x}}{1 + e^{8 x}}$ .

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

Step 1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{1 - e^{8 x}}{1 + e^{8 x}}$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{d}{d x} [\frac{1 - e^{8 x}}{1 + e^{8 x}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - e^{8 x}$ and $g (x) = 1 + e^{8 x}$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - e^{8 x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- e^{8 x}]$ .

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

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{(1 + e^{8 x}) (0 + \frac{d}{d x} [- e^{8 x}]) - (1 - e^{8 x}) \frac{d}{d x} [1 + e^{8 x}]}{{(1 + e^{8 x})}^{2}}$

Step 3.3

Add $0$ and $\frac{d}{d x} [- e^{8 x}]$ .

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

Step 3.4

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Multiply $8$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Multiply $- 8$ by $1$ .

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

Step 5.5

By the Sum Rule, the derivative of $1 + e^{8 x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [e^{8 x}]$ .

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

Step 5.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{(1 + e^{8 x}) (- 8 e^{8 x}) - (1 - e^{8 x}) (0 + \frac{d}{d x} [e^{8 x}])}{{(1 + e^{8 x})}^{2}}$

Step 5.7

Add $0$ and $\frac{d}{d x} [e^{8 x}]$ .

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

Step 6

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

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Step 6.1

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

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{(1 + e^{8 x}) (- 8 e^{8 x}) - (1 - e^{8 x}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [8 x])}{{(1 + e^{8 x})}^{2}}$

Step 6.2

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

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{(1 + e^{8 x}) (- 8 e^{8 x}) - (1 - e^{8 x}) (e^{u_{3}} \frac{d}{d x} [8 x])}{{(1 + e^{8 x})}^{2}}$

Step 6.3

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

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

Step 7

Differentiate.

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Step 7.1

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

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

Step 7.2

Multiply $8$ by $- 1$ .

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

Step 7.3

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

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

Step 7.4

Multiply $e^{8 x}$ by $1$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{(1 + e^{8 x}) (- 8 e^{8 x}) - 8 (1 - e^{8 x}) e^{8 x}}{{(1 + e^{8 x})}^{2}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{1 (- 8 e^{8 x}) + e^{8 x} (- 8 e^{8 x}) - 8 (1 - e^{8 x}) e^{8 x}}{{(1 + e^{8 x})}^{2}}$

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Simplify the numerator.

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Step 8.4.1

Simplify each term.

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Step 8.4.1.1

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

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

Step 8.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.4.1.3

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

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Step 8.4.1.3.1

Move $e^{8 x}$ .

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

Step 8.4.1.3.2

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

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

Step 8.4.1.3.3

Add $8 x$ and $8 x$ .

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

Step 8.4.1.4

Multiply $- 8$ by $1$ .

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

Step 8.4.1.5

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

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Step 8.4.1.5.1

Move $e^{8 x}$ .

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

Step 8.4.1.5.2

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

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

Step 8.4.1.5.3

Add $8 x$ and $8 x$ .

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

Step 8.4.1.6

Multiply $- 1$ by $- 8$ .

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

Step 8.4.2

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

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Step 8.4.2.1

Add $- 8 e^{16 x}$ and $8 e^{16 x}$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{- 8 e^{8 x} + 0 - 8 e^{8 x}}{{(1 + e^{8 x})}^{2}}$

Step 8.4.2.2

Add $- 8 e^{8 x}$ and $0$ .

$- \sin (\frac{1 - e^{8 x}}{1 + e^{8 x}}) \frac{- 8 e^{8 x} - 8 e^{8 x}}{{(1 + e^{8 x})}^{2}}$

Step 8.4.3

Subtract $8 e^{8 x}$ from $- 8 e^{8 x}$ .

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

Step 8.5

Move the negative in front of the fraction.

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

Step 9

Simplify.

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Step 9.1

Combine terms.

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Step 9.1.1

Multiply $- 1$ by $- 1$ .

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

Step 9.1.2

Multiply $\sin (\frac{1 - e^{8 x}}{1 + e^{8 x}})$ by $1$ .

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

Step 9.1.3

Combine $\sin (\frac{1 - e^{8 x}}{1 + e^{8 x}})$ and $\frac{16 e^{8 x}}{{(1 + e^{8 x})}^{2}}$ .

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

Step 9.2

Reorder terms.

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